This transaction would be in the mempool until could be included in a block.
The transaction won't be in most node's mempools (note that there is no "the mempool"). Most nodes will not accept a transaction whose locktime has not yet passed.
Can a "well paid" transaction hang around in the mempool for months/years?
No. Transactions that have been in a node's mempool are automatically evicted after a certain amount of time (default is 2 weeks). You could rebroadcast the transaction routinely, but in your case, that won't help.
How can it be checked if trown out of mempool?
Not a concern because it won't be accepted to the mempool in the first place.
Is the locktime strictly enforced?
Yes. If a block contains a transaction that has a lock time that has not yet passed, it will be invalid.
Is the locktime value unmodifiable?
Yes. Your digital signature protects the locktime, just like all other parts of the transaction. Changing the locktime would make the signature invalid thus making the entire transaction invalid.
I assume the future timelocked mempool transaction shouldn't prevent spending the funds without current locktime?
Yes, You can spend the same output in a different transaction thus invalidating the locktimed transaction.