What happens if a miner publishes a block which contains an invalid transaction? What happens if a miner (accidentally or maliciously) publishes a block with correct proof of works, which contains an invalid transaction (e.g., a double spending one)? Is the block ignored by the network?
A block with an invalid transaction is an invalid block, regardless of the proof of work.

When it is broadcast, other full nodes will ignore it as spam, and typically disconnect from the node that sent it.  A bad block is unlikely to propagate very far on the network.

When other miners see it, they will also ignore it.  If they mine on top of it, they will be wasting their hashing power.  This also means that the miner who did the proof of work on the invalid block has wasted their own hashing power.  This is why it's important that miners validate the transactions they include in blocks.