Does every miner verify blocks that are already in the Blockchain Once a miner adds a block to the chain, does every other miner have to check that block to make sure that its transactions don't violate any rules? For instance, if a miner added a block with an invalid transaction, is every other miner expected to check that block (and perhaps compare each transaction in it with every other transaction in every other block, to prevent, say, double spending)?
I'd advise you to use the search function a bit more, as you are asking a lot of questions which have already been answered.

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.

What happens if a miner publishes a block which contains an invalid transaction?