I missed this thread on its first go-round, but now that it has been “bumped”…
a) As stated, above, laser cutting plastic is legal, but…
Laser cutting Polycarbonate isn’t the best idea. It releases some very nasty gases that can damage your laser cutter. Epilog (one major manufacturer of laser cutters) goes so far as to say that cutting polycarb with their laser will void the warranty. (Yes, I know… voiding the warranty is part of the fun, but just not with my laser, please.) Yes, it will cut - not nicely, but it will cut. Yes, you can use it. But no, I don’t recommend it. Perhaps PETG, as far as the transparent polymers go, will be a better bet. Here’s a few other materials that might work… just remember to use the “non-shattering” ones for VEX. https://www.pololu.com/docs/0J24/3
b) See above
c) Feel free to hand cut whatever you want. Polycarb cuts beautifully in a water jet, and isn’t bad on a CNC router or mill, either. Cut away. Use a light saber if you’ve got one.
And let me add:
If you’re only cutting a Logo, for use as a non-functional decoration (and make sure you read the definition of non-functional) you can use acrylic, cardboard, hardboard or dilithium crystals. Well, okay, the crystals might be prohibited under safety rules. But non functional decorations do not count to your “plastic sheet” allowance. So long as they are 100% non-functional decorations. Heck, you can even 3D print your non-functional decorations.
As for using the laser to cut the sheet down to the correct thickness, the statement, above, that you can’t use the laser to cut through the piece is obvious… but you can use the laser to ablate the surface until you have material of the correct thickness. This strikes me as a slightly ridiculous approach, particularly when sheet of the correct thickness is commercially available… but in the event of the zombie apocalypse where you are trapped in a building and absolutely need a sheet of 0.070" thick plastic to avoid having your brain eaten. Yeah. It might work… might. That’s really not a great way to do it unless the zombies are coming for you, though. If you’re in the USA, then McMaster Carr is just as “local” as Amazon https://www.mcmaster.com/#=17uezdu
And as for the immediately preceeding comment about using a saw… you aren’t wrong, but you haven’t used a laser cutter, have you? They are beautiful things.