How is a miner prevented from adding transactions without signatures to their block?
No, you're not the only one who checks signatures.

Before a block is submitted, every signature / transaction is checked/verified by other nodes.

I believe the SHA256 hash is the first thing which they check, which contains all/most headers for the block.Then every tranasaction hash +/signature is checked.

If they can't confirm a transaction hash / block hash i believe your block will be rejected.

This might help you out a bit, -> https://en.bitcoin.it/wiki/Protocol_rules#Explanation_of_Some_Rules