Copper Coil and Electromagnetic Induction
Magnetic induction doesn’t need any thickness of coil (presumably you mean wire or conductor thickness) to work. Radio waves rely on electric fields and magnetic fields interacting via induction and there’s no wires involved hence why it’s called “wireless”.

In fact, a conducting wire can be a bit of a hindrance due to induced eddy currents taking away a bit of the energy and converting it to heat so, the thinner the wire the better. But if you need to produce decent amounts of current in “the secondary” then you need wire that is bigger.

If when you say the “thickness of the coil” you mean the coil diameter then the bigger the diameter the more induced voltage you are likely to get so induction can be improved because a larger coil diameter can receive more flux due to it encompassing a larger area.

Copper is commonly used because it is a good conductor; whether that is due to more or less electrons in its outer shell you’ll have to ask a physicist but there will be plenty of google sites that will tell you.

Your third question is answered in my first paragraph but, just in case, mixing wire diameter isn’t a big show stopper.