The church, train station and shops being locked is all explained over the course of the story. As for Lyx, make sure that you've conquered at least the Bridge and the Apartments and then do the night time event at the Bridge (where you try to scout the church).
After that, Claire will appear. Then Lyx will reappear when you sleep, and you simply follow your lead from there.
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Sleep. The triggers for Lyx are super simple:
First Appearance: Sleep after being in a fight.
Second appearance: Sleep after getting Claire.
The variable calls are extremely basic so there's no chance of the code getting confused or failing.
So long as you're using an official version, not applying translation patches or mods, and not letting anyone mess with your variables, everything will work correctly.