Bitcoin Core does not interface with ASIC miners directly. Bitcoin Core no longer contains efficient mining code because mining with a CPU is pointless and just a waste of energy. The implementation of SHA256 is just used for the computation of the hashes used all over Bitcoin, not for any mining purposes.
I'm curious about interfacing with a rig. Can you tell me if the rigs being sold these days interface directly to the bitcoin network or whether they interface with a computer which interfaces with the bitcoin network. In the latter case, what software is available to run on the computer, and is the interface between the computer and the rig standardized, or does the software need to be customized for each model of rig out there?
Asics these days can either connect to a mining pool directly or connect to a driver running on a computer which connects to the pool. Either way, they usually run some version of cgminer or bfgminer which are software that can communicate with these devices. These software also speak the stratum protocol used by pools and the getblocktemplate protocol used by Bitcoin Core (and some other pools).
Would there be any serious interest in publishing my code as part of Bitcoin Core, and would it be possible to license this particular module, or perhaps publish it as shareware, and earn a little money from it?
No, not as a part of Bitcoin Core.