It looks to me like the way the "clocks and rods" idea has been introduced to you is more complicated than necessary, and this has caused you to have trouble understanding it. It is in fact very simple and down to earth, although it can be used for some subtle purposes.
It's sad to see what a state you are in after reading your "introductory text" which failed in showing how utterly awesome lattices of clocks and rods are. You sound unenthusiastic about relativity, and even a little bit depressed. My heart goes out to you. It's so sad that something as, if may say so, beautiful, as the clocks and rods lattice thought experiment was put across to you in a way that failed to fire your imagination. The reason is probably that the original use of the clocks and rods lattice thought experiment was by Albert Einstein in 1905, and his purpose was not to explain anything, but rather to make his theory as rigorous as possible. And, as happens all too often in physics (and even more so in math) students (even beginners) are talked to as if they were miniature professors, full of skepticism and therefore in need of proof. What follows is based on the idea that you want to understand relativity, not see it proved or made more rigorous than it already is.
First, you need to understand what "synchronized" means here. The use of high tech light pulses and so on can make it seem like this word is being used in some special, different sense in relativity. In fact, the word has its everyday meaning.
To understand this, consider when a screen gang of criminals sychronize their watches, as part of preparing to rob a bank, say. One way this could happen is for the gang leader to visit each of his underlings and adjust the underling's watch (or supervise the underling while he does so) so that it is displaying the same time as his watch (which we could say is the "master watch"), and then warns the underling to use only this watch, and not adjust the time on it, until after the robbery has been completed. Once the gang leader has visited all his underlings, it can be said that they have successfully completed the task known as "synchronizing watches" in every movie.
The clocks in the rods and clocks lattice are synchronized in the exact same sense of the word. There's no need to use light pulses or radar or radio signals to do it. You could carry a master clock with you and adjust each clock in the lattice, one by one, by visiting it and putting the master clock and the lattice clock next to each other and adjusting the time on the lattice clock so that it matches that of the master clock.
The point of the light or radio pulses is that it is a quick and convenient way for a physicist to do it in practice. It has nothing to do with the meaning of "synchronized" or anything to do with the fundamental nature of the thought experiment. These are ordinary clocks, telling ordinary time, synchronized in the ordinary way, which just means that they all display the same time once the synchronizing process is completed.
It does not mean that the sound of the ticks reaches your ears simultaneously. Clocks at the far end will seem to tick just after clocks at the near end. This is just signal delay.
Likewise, if the lattice is big enough or you have a video camera that high speed enough, the finite speed of light means that you will see the second hand of a clock nearby tick over before you see the second hand of a faraway clock tick over. This is just signal delay again. In one case it's due to sound taking time to cross space, and in the other due to light taking (approximately a million times less) time to cross space, and does not change the fact that the clocks are synchronized. In fact, if you did see all the clocks ticking over at exactly the same time, despite some being near and others far, that would show that they were not synchronized.
The way the lattice is used to make measurements is that you have a technician manning each clock. Each clock is attached to the point where six rods meet at right angles. We can call this a node in the lattice. It has a definite location, and the time is displayed there by a clock, and there is exactly one technician there, ready to make measurements. The clock doesn't necessarily "join" (as you put it) the rods to each other. To do that it would have to have zero size, or it would add to length of the rods.
Now imagine a rocket flying through the lattice. When the front of the rocket passes a node in the lattice, the technician at that node notes the time displayed by his clock, and that the front of the rocket passed at that time. A little bit later the back of the rocket passes, and the technician notes the fact, and notes the time that it happened, in the same way. Then he writes a report, seals it in an envelope and mails it to the bookkeeper, who, along with all the envelopes from all the other technicians, opens the envelope, and makes a table containing all the data, and from that is able to work out what the position was of the the front of the rocket at what time, and likewise for the back of the rocket.
There's no advantage to stopping any of the clocks, not even in an automated lattice.
Note that when you wrote in your question,"If any event occurs, it stops the clock at which the event occurs and makes a mark on the rod." you are essentially referring to an automated version of the lattice where when, say, the front of the rocket passes a node it stops the clock belonging to that node, and makes a mark on the rod (it would work equally well if it made a mark on the clock). The mark might be red because the dye on the front of the rocket is red, or blue because the back or the rocket is blue, although in this case, since the clock, once stopped, cannot be used to make a second time measurement, the time that the blue back of the rocket passes can't be recorded by that node). If you are only interested in the front of the rocket, marking the rod adds nothing, because the stopped clock has a fixed and known or measurable position in the lattice. In fact, it's confusing to say that "the rod" (which or the six rods that meet there?) gets a mark. Where the rod? If there is to be a mark, it needs to be at the exact same location as the clock is at, and like I said, why not just mark the clock in that case. If that's what your introductory text says, I suggest you look at some other texts as well.
The bookkeeper can use the data he receives in the envelopes to calculate the length of the rocket, as well as it's velocity, at any time that it was inside the lattice. If the rocket is accelating, that can be calculated, too, with the same when and where precision. All this from those simple reports sent in by those technicians, that only specified what was where and when. They didn't themselves try to measure the speed, direction, or acceleration.
Even more fun is when the front and the back of the rocket each have a clock attached to it. Then the technician can add to his report what each moving clock is displaying as it passes him.
You might be wondering about the time it takes for the light to get from the moving clock to the lattice technician's eye. The idea in this thought experiment is that the measurement of the moving clock's displayed time is done at the node, in effect by a process involving direct contact, something like printing, or brass rubbing (except instantaneos). The clocks are imagined as being points, or of negligible size. The point of this is to avoid having to consider what the effects of signals, especially signal delay might be.
Many of the misunderstandings of relativity are due to misunderstanding the role of signal delay, so it makes sense to take them out of the picture when relativity is being introduced. Using the lattice achieves this by allowing all measurements to be made at each event, with negligible signal delay. The light might travel a meter or two, say, from the front of the rocket to the detection device in a typical lattice, which is near as darn at it nothing. It's negligible. There's no need to completely eliminate signal delay. On the other hand, it's worth noting that it is possible to eliminate all signals from the thought experiment, and this could be helpful for some students. This is done by using direct contact.
There are many ways that you could have the clock(s) on a rocket moving at high speed read by direct contact with a lattice that is at rest. One way would be to have the time on the clock converted to a binary number, and then indicate that number by a row of pins on a flat rod. Pin out means a one and and pin in means a zero. The pins that are out would leave marks on any rod they struck. Or the pins could electrically charged in opposite ways, or magnetized in opposite ways. As the row of pins passes a node the pattern of electric charge or magnetism could be recorded automatically.
You may be thinking that this lattice is a lot like a frame of reference, and you'd be right. It is in effect a thought experiment embodiment of a frame of reference. Just as you can have moving, and even accelerated frames of reference, you can have your lattice do those things, too. There's very little difference in practice between a lattice of clocks and rods (and technicians if it's not automated) and a frame of reference. I guess a frame of reference would be the mathematical abstraction you get from the lattice's behavior.
Anyway, the lattice is the ideal way to introduce relativity, and the beauty of it is that the way it works is entirely Newtonian, and you can teach it and use it as part of teaching Newtonian mechanics. The process of construction and calibration is just common sense.
It's a way to clearly explain, and rigorously define, what a frame of reference is. Frames of reference, relative motion, and even moving and accelerated frames of reference all have meaning in Newtonian physics. So mastering the lattice can be done without touching relativity, but it creates the ideal solid understanding (of Newtonian physics) needed for trouble free assimilation of Einstein's special theory of relativity.
In a nutshell, if you have doubts about the behavior of signals, in effect you do not trust your own eyes. What do you do in that case? Feel your way. And that's exactly what the lattice is for. The way it works is tactile. Events within it are in a sense "felt" (detected by direct contact).
At normal "low" nonrelativistic speeds, that rocket will behave in a normal Newtonian way. It's length won't change nonnegligibly, for example. It's clocks will tick at a constant rate and so on. All this can be verified using the lattice (if you built one, or an equivalent piece of kit).
When you repeat the experiments with relativistic speeds, e.g. half the speed of light, the lattice will show you a couple of interesting things. First, the length of the rocket as measured by the lattice, which means within the frame of reference of the lattice which is at rest in our example would be less. This means that for us, at rest, the rocket really is shorter. It's not some sort of optical illusion, which is a common way relativity is explained, some sort of trick of the light that becomes real for unclear reasons. When the rocket is travelling at half the speed of light, there are a lot of interesting "tricks of the light". For example, the light from the rocket will seem to show the rocket flying diagonally ("rotated") but this isn't what is actually happening, and the lattice measurements would show that it's orientation hasn't changed for anyone in any frame of reference. It's just that the light from the far side the rocket arrives later, making it look rotated, but only look rotated. So as you can see (no pun intended), there's a lot be gained from taking signals out of the picture.
The clocks and rods lattice is one of the most underrated things in physics education. I can't believe how few articles devoted to explaining what a frame of reference make any mention of it. It's absolutely crucial (no pun intended).
A great thing about the lattice is that you can see first what Newtonian physics and relatistic physics have in common, and later what they don't have in common.
The exact same lattice still serves even when rotating and even when spacetime is not flat, for example near a black hole. It just gets a bit bent out of shape, so to speak. So it will still be useful even when you start studing general relativity.
You can also have one 3D lattice moving through another in a thought experiment. In a lab you can a have a "2D lattice", or more likely, a "1D lattice", that is to say, a sheet or more likely a line of clocks and rods moving over another, each one taking readings of its own and the other's clocks and their positions. This helps with visualizing the Lorentz transformation.
The clocks can be candles that burn down slowly as time passes for easy visualization in a thought experiment.
The alert reader will be wondering about the relativistic effects on the master clock when it is transported around the lattice when the clocks are being synchronized. The answer to that is threefold. First, as long as the lattice can fit into a medium sized city, and the master clock is moved about no faster than an automobile can carry it, the time lost by it due to slowing of the master clock due to its journey aroung the lattice will not even be detectable, let alone nonneglibible. Second, the effects of relativity are huge and there's really no need to even synchonize the clocks. You can just use some clocks that are all showing the current time. The use of pulses of light to synchronize the clocks is liable to mislead the student into thinking that superprecision is needed in sychronizing the clocks and that therefore the change in length of the rocket are slight, and hard to detect. Nothing could be further from the truth. The rocket is half as long as it was at rest. That's not a subtle effect. Third, even if the lattice is so big that moving the master clock around it could conceivably cause it to lose a detectable amount of time, you can reduce that time to as little as you want simply by moving it slowly enough. By taking twice as long to visit all the clocks you cut the total time lost by the clock by a factor of about two. So a large distance is no problem if travelled slowly enough.
I hope that helps. Many people learn of the rods and clocks lattice from an awesome book by Lewis Epstein called "Relativity Visualized". One reviewer called it "the gold nugget of the relativity books". If you liked this answer you'll love Relativity Visualized.
Here's a screenshot of part of Wikipedia article called "Einstein synchronization" included as evidence that what I wrote above is correct. I used a screen shot because copy pasting didn't work with the symbols. Sorry it's so small. Windows has a screen "magnifier" that may help (shortcut: press the equals sign key while holding down one of the Windows icon keys) or try right clicking in the image and selecting "open image in new tab" from the pop up menu.