And here we delve into something called Pop Culture Pagansim. Or at least how I see it. Yes, you can use that wand for actual occult purposes. Just keep all the actual fake Hollywood stuff out of the equation and you're good. Just make sure you cleanse it and charge it and you should be good to go.
You certainly can, but wands are often chosen due to the proporties of the wood. Plastic wands or wands that are of an unknown wood are more for roleplay than actual craft work.
It is far easier and cheaper to go out and find a piece of wood that draws you to it, or find out what kind of wood you would like to make your wand out of and find a piece of fallen (non rotten) wood.
You can then add carvings, crystals, symbols etc, that you would like to the wand. You can stain it, shape it ect to your liking.