CATastrophe

I can't be sure since I haven't tried it, but it makes sense. If you're really worried, Dorothea is easy to recruit anyway (Byleth naturally gets a lot of charm and as long as you have a battalion equipped, you should eventually have a high enough authority level).

It seems that way. Maybe you can get the relics other ways if you recruit the requisite characters after you play the paralogue, but I don't know. I'm not sure about after the skip, but pre-skip it seems the only thing you get from Dorothea is a goddess ring, which you can get other ways. Thyrsus is from Lorenz's paralogue.

I don't think that's true. I went BL and recruited Dorothea and got both the goddess ring and Luin from their paralogue pre-skip. In fact, I've gotten a ton of relics and special weapons from pre-skip paralogues. (So far, Lance of Ruin, Luin, Spear of Assal, Thyrsus, Aegis shield, and the whatever the Cethleann staff is called. I'm working on Hilda's now.) It is indeed probably because they didn't have Ingrid that they didn't receive her Relic.