If I'm understanding segwit correctly, miners have to verify signatures in order to include a transaction into a block that they are mining. However, miners are not required to verify signatures in order to decide that a block a different miner mined is valid. Is this correct?
Miners can do what they want to construct blocks, including creating blocks with invalid signatures. They can even try construct blocks by just creating random data, and hoping it's valid.
However, blocks without valid signatures (or other validity problems) will not be accepted by the network. All full nodes verify all signatures in all blocks, so if a miner constructs a block with an invalid signature, they've just wasted a enormous amount of effort to produce something everyone will just ignore, and forgone the ability to be paid for that block. The same is true for building on top of an invalid block; thus miners are incentivized to validate the blocks they build on top of too - but they don't have if they like losing money.
None of this has anything to do with segwit. Segwit moved signatures to a different place, where they don't affect txids, and can be stripped cleanly so pre-segwit nodes can ignore them, but doesn't mean nodes don't validate anything less (except old nodes, which don't care about the segregated area).
It sounds to me like most do still verify though?
Definitely.
Are signatures discarded eventually?
Not in any current full node implementation that I know of.
Bitcoin Core supports pruning old block data, which deletes both witness and non-witness data equally. That also means other nodes can't synchronize off a pruned node anymore.
In theory it's possible to implement a pruning mode that deletes just the witnesses. That would also prevent other (post-segwit) nodes from synchronizing off them, but they could still serve some light clients. This is not implemented.
Should a mining pool temporarily gain 51% control of the blockchain, can they insert transactions, and not include the segwit for a block or two because, by the time other miners are allowed to mine, those signatures will have expired?
I don't understand what you mean here, but signatures don't "expire".
