Hi,
I don't give nothing to nobody - Double, triple, etc. negatives are a feature of some dialects. That's pretty common, but I'm not sure if it is the standard form in those dialects or if it is just a possible structure. In other words, I'm not sure if in those dialects "I don't give anything to anybody" or "I don't know anything" would be uncommon compared to "I didn't give nothing to nobody" or "I don't know nothing".
Those forms are only acceptable and used in certain dialects.
Me neither / Me either - Both should be used in American English, but I think "Me either" is not very common compared to "Me neither". Not sure about this, though. It depends.
Now wait for someone else's reply.