Does it make sense to create a spacetime diagram in respect to another velocity besides the speed of light? I'm trying to understand the application of special relativity into things where it's velocity might be none regular units. Just for argument sake, let's say that the distance is measured in some unit called "sticks" and time is measured in hours. And these sticks have no way of being converted to meters.
If the object is moving at 100 sticks/hour and it's changing direction left and right and is at different positions, is there anything wrong with creating a space time diagram where time being sticks/hour instead of speed of light? This way we have the same units for time and x axes so that I can perform some relativity calculations?
 A: In your spacetime diagram, you can't have a light ray go at an arbitrary angle.
Before you can make use of your diagram, you'd need to go set up an experiment, and measure the speed of light in sticks per hour. Once you obtain the value, you'd go that number of sticks to the right, and one hour up, and mark that point. Then you'd connect the origin with the marked point to obtain the line representing the speed of light.

Or, if you actually have the stick as some sort of a reference etalon, just go and measure how many sticks there are in a meter (in the literal sense: take something that's 1m in length, and measure it using your stick as a unit) - and there's your meter-to-stick conversion factor.
P.S. BTW, this is why one of the axes typically has "unusual" units (or non-uniform scaling); the speed of light is so great that if you used a uniform grid with everyday units, the diagram would be impractical:

A: In spacetime diagrams, the time axis is measured in meters, with the speed of light being the conversion factor.
If you do your computation in sticks, then your $x$ axis is sticks, and your time axis is in hours $\times$ (the speed of light in sticks per hour), which is also sticks.  The speed of light is a special velocity in their system, which is special independently of the system of units.
