Sacred Haven of Debauchery

The production of resources suddenly and sharply decreased; it is impossible to build anything, and the planets in the panel on the right have stopped being displayed.

1. This means that the sector is too "isolated" and keeps most of the resources it produces for itself. This needs to be adjusted in the sector panel, and a reasonable tax rate should be set for the sector.
2. Of course not! Sectors have indirect control. And this is a good thing in some ways: it reduces micromanagement when there are many planets; if you have to build everything manually, you will get tired. If you need to build something that the AI cannot do on its own, you can temporarily exclude the planet from the sector and then include it again (but the influence will be spent).
3. See 2. But it should be added that spaceports are still available (both for the player and for the sector AI), and this is a huge plus.

By the way, yes, I noticed that the AI does not build anything in the sector, no matter how many minerals you give it.

If this is an AI bug, that's bad... Mine was building and consuming resources quite well. But it didn't know how to improve administration, and that's a big problem.

I had to build the planets' infrastructure myself (fortunately, it doesn't cancel your long-term construction queue).

Is it just me, or is all the infrastructure being built? Give it a priority, and it will happily fulfill it! Sometimes it even overfulfills, building buildings in a not very optimal way across sectors.)
Also, if all the inhabitants of the planet are busy, it may not build anything, but that's logical.