Is the minimum fee considered when a full P2P node validates a Tx? So if the core devs change the minimum fee schedule, lowering the maximum "free" tx from 10,000 to 1,000 bytes - how will that not cause a hard fork? Or at least prevent nodes running old software from propagating a new Tx?

https://github.com/bitcoin/bitcoin/blob/master/src/main.cpp

I could be reading this wrong, but looks to me like the minimum fee schedule is part of the "standard tx" test, and so nodes with different fee schedules wouldn't be able to send all tx's between each other.
A bitcoin network fork occur when a block generated by one node is not accepted by a percentage of nodes, those nodes will split and continue working on its own.

This change will not cause a fork, it will cause that some transactions will be discarded by some nodes, older nodes will never include it on a block but newer nodes will and older nodes will accept a block with transactions different transaction fees as those are not checked for block validity.

This has already happened, pools can discard transactions without fees or ask for higher fees than usual for example and the block will be valid and included in the blockchain without problems.