Is it possible to remove tokens from dead projects/tokens that are totally useless and worth $0 from my wallet? Do I just send them to a random address? I don't want to burden anyone else with those shitcoins 
You can't remove token from wallet, I think you just need transfer your tokens that still have value to new wallet, and leave the dead project token or worthless token from your wallet
There is no point in creating a new wallet every time you get a token with a zero price. If you participate in bounties and airdrops, you will have many more such tokens. Just learn not to notice them, especially since it is possible to make them invisible.