Electronic – Electrified Vertical mounted rotary joint questions

Your while loop logic is incompatible with the conditions inside it.

Your while loop is saying:

as long as BOTH wheel encoders read less than clicks, execute the code inside

and your if statements are saying:

if any of the wheel encoders is greater or equal to clicks, stop the corresponding motor

This is generally false except for the rare condition where one of the encoders happen to increment between the arduino evaluating the while and the if statements. In order words just about never.

So the first thing you need to do is fix your logic. Then you need to investigate more sophisticated means of keeping the wheels in sync, such as computing an error term from the different in ticks, and employing a PID controller algorithm to adjust the speed of both motors via analogWrite.

Your question is a general electrical design one BUT is couched in terms of robotics. As such it is likely to be deemed inappropriate to this group and transferred or closed. (Hover over the robotics tag and see what it says.) Likely choices are move it to Robotics page or reword. If you are interested in rewording it as a general electrical question it may be acceptable here. (NB - I don't make the rules - I am just an "observer".).

While there are theoretical answers for many of your queries, for the parts that you have or will have soon, a good way to answer them is by experimentation. Think in each case about whether you are liable to damage something by experiment and if not, try it. In many cases you can be fairly safe by use of fuses and a small amount of resistance (or some long thin wires at first) to limit peak power and current when things take an unexpected turn.

Running on 5V seems worth trying. Power dissipation under stall MAY be higher than desired but intelligent playing should be safe enough.

Wire sizes? Wire tables abound on net with voltage drop per metre at given current and/or Ohms/metre. But trying a metre or so and measuring the voltage drop can be very instructive. While doing this, make sure any connectors allow access to the wire just either side of the connector - measuring voltage drop across connector pins can be'interesting'.

Mixing gauges in series? What will that do? Essentially it's a voltage drop issue with heat of interest in extreme cases.

I see screw down connectors (chocolate blocks) in that photo.
Check voltage drop across the connector as above. Twisting wires together helps. Cheaper connectors (probably as per photo) tend to chew up wires when screwed up and opened several times. If you wish to do this you can get connector blocks with clamps (dearer) or use ferrules on cable ends or ... .

Note that (perhaps surprisingly till you know why) one thing that you must NOT do is fully solder wire ends before inserting in connector blocks. What happens is that the wire bundle initially resists the screw pressure but, over time, the solder "creeps" and the connector loosens its grip. If you solder wires together and then use a connector block just to retain the wires in place you MAY not get this problem - as long as the wires stay soldered together.
For separate wires it is permissible to solder tin the very outer end to keep the wire from fraying.