Project Squirrel

[Bodge Job]

Sounds like a good first robot plan, spinners may be better to avoid for first robots though as there much more dangerous and expensive, (can only be run at certain events as well). For good design ideas have a look at other robots and watch out for how there made and there main advantages and failure points (eg do they always land on the wheels, or get stuck on their back/side) and keep these in mind when designing yours.

Battery wise, what kind of batterys are you talking? (lipo's, nimhs or Lead acid etc) if your new to things, nimhs are a better choice as there cheaper and safer. Lipos are lighter for the same energy so are slightly better, but dangerous if damaged or miss used (can catch fire) so require special equipment and have extra rules, and will end up coming much more expensive so I'd avoided using those at first unless you have prior experience. lead acid batterys (sla) are generally too heavy and only supply a low current so I'd avoid them altogether for feathers. (you may know all this already, but wasn't sure)
From the cell count your mentioning, you seem to be referring to Lipos (3.7 volt per cell). so a 4s is probably a good bet for 12v drills, a 6 cell would be 22.2 volts so would probably kill your motors and esc very fast so I wouldn't use that. Most motors can survive a 50% overvolt eg a 12v motor could survive being used at 18v, but would massively shorten it's life, (currents would be much higher, so every thing will get much hotter) some motors can't cope with any overvolting though so be careful.
If using nimhs, there 1.2 volts a cell, so 10-14 cells is a good bet (could use two identical 7s packs in series to achieve a 16.8 volt pack.
always make sure to keep below your ESC's max voltage, keep in mind lipos peak at 4.2 volts per cell when fully charged.

Plastic related questions:
https://www.directplastics.co.uk is where I get my plastic from, chopping boards maybe be cheaper but are often made from LDPE, which tends to crack/split when hit by spinners. Thickness is difficult to judge (depends a lot on shape and how it's used etc) 5-6mm will generally be enough for your top and bottom (little hitter only managed to make it though my 6mm top once), not sure how that survives against crushers though. For side armour some robots rely on thinner plastic (I think some bits of explosion are 5-7mm, but spinners can often make it through that, 10mm can be enough, but still a tad on the thin side. I've found 15mm+ is a safe bet, and is thick enough to be screwed into end on, so you wouldn't need angle section or mounts to hold the plastic together (so thicker plastic can save weight if used in the right way)

Some chunky countersunk wood screws works well in this plastic, and you can used threaded metal inserts with bolts if you want a longer life ( good for commonly removed parts like a baseplate). Hdpe is too soft to be threaded for normal bolts though, so avoid that.

With weapon mounting, Hdpe is a bit soft for spinner shafts to run directly onto , so it will warp over time. It may be fine for axes though, provided you can keep the chain tensioned. I believe some People have had success with spinner shafts going directly into Nylon 6 (tougher, but more expensive plastic) haven't had any experience with that my self though)
Bearing wise, some people swear by bearings (less friction) , others by bushings (less to break) , I don't have enough experience with spinners to tell you either way. If your going for bearings I've found https://www.technobotsonline.com is pretty good, others companies like bearing boys also have a large selection. Not bought bushings before so don't know of a good source.
For wires, I normally use 2.5mm, but that's pretty over kill (it's what I have laying around) I think 1.5mm is enough for drill motors but probably to thin for battery wires so go for somewhere in between. If you want a cheap source of wire I've found buying a cheap 13amp mains extension lead and chopping a chunk out of that, is cheaper than buying rolls of wire at hardware stores ( you can then wire the plug back on so the extension lead still works, be careful doing that if your not use to wiring mains stuff though).

Sorry, someone else will have to answer the rest of your questions as I'm out of time!