Can a miner just download the blockchain once and then delete it (don't have a full node)? I am wondering if all bitcoin miners need to be full nodes. Or is it possible for a miner to just download all the blockchain once, do all necessary computations to start mining and then delete the complete blockchain and only work on the "last blocks". I.e can a miner don't be a full node?

This is not a duplicate question.
If miners delete the blocks data, someone can fool them by sending a transaction that spends money it shouldn't be able to. In principle, some of them could do that and trust the other miners to validate transactions, but the more people do that, the more the whole network is vulnerable to fraud.

The same can be said for non-miners. If you don't run a full node, you're not actually running Bitcoin, but trusting people that run it to verify transactions for you.