How does adding the equivalent of APN scripts work for VScode? Do they go in the main extension, or perhaps we need to have our own extension for custom scripts?

I am unfamiliar with APN Scripts, I don't think we use them.

I imagine, depending on the script, a Task in tasks.json maybe a good solution akin to the compile task.

I'd guess you could probably also attach the script to the end of compil.ps1 and have them fire off automatically following a compile (if the compiler succeeds).