Addressing the 3rd question, the clocks' frequencies are the same when each is viewed in its own rest frame. When a clock is viewed in a moving frame, that's when its frequency is changed. You record the ticks and their locations (which are different since the clock is moving) and you discover that the time between ticks is now longer.
Oh, and addressing the second question, no. This has to do with the meaning of temperature. A temperature is function telling what the occupation-number is for each energy-level of the system. What exactly are the thingies that occupy each energy level can be very un-intuitive; it isn't always just whole atoms or whole particles. So, we'll skip that. But the main point is that for each energy, there is a number, and for each temperature there is a function.
OK, so what happens when the actual occupations don't match any temperature function? The distribution is said to be non-thermal.
In a cesium clock, this is the case. There is a temperature associated with the atoms before you start measuring, but once you shine this wave on them, the one associated with the hyper-fine transition, then the number for that one energy level goes 'way up. So we have a mostly thermal shape with one wrong energy-level. So, it's easy to see why someone would call that thermal and name a temperature. You just omit the one odd energy-level.
If you do that for absolute zero, you would still be able to have a clock; the clock would still run. You just have to understand what is meant.