1. Inversion vs. Est-ce que
Yes, it is true that inversion is considered more formal than ‘est-ce que’, but there are many fixed expressions where the preference is to use the inversion (just like your example - quelle heure est-il?). Sometimes it is simply a matter of being quicker to say, or sounding better.
2. Preferred Format/Structure - it's up to you
I'm not a huge fan of the inversion, because it can sound too formal, however that completely on how you phrase the question as well. There are a myriad of ways you can do this, so I'm going to lay out a few below using your example.
- Anne s'est sentie mal à l’aise après s’être trompée en saluant quelqu’un. (Statement)
- Comment Anne s’est-elle sentie après s’être trompée en saluant quelqu’un? (Inversion)
- Comment est-ce qu’Anne s’est sentie après s’être trompée en saluant quelqu’un? (Est-ce que)
- Annie s'est sentie comment après s’être trompée en saluant quelqu’un? (Interrogative last)
- Elle s'est sentie comment, Anne, après s’être trompée en saluant quelqu’un? (Interrogative last and pronoun shifted)
There are more options, but we are going to start sounding like Yoda if we start down that route. My preference is for the last one. In this case, it is more natural for me to throw the question word in at the end, splitting the two clauses and allowing to you put emphasis on the central question, then providing the context afterwards. Even with simple, single-clause sentences I would often just throw the question word in at the end.
If I understand your question correctly, I don't think in French we make such a big distinction between the two forms (est-ce que and inversion) in terms of literal meaning, like you pointed out with the English (How is it that Anne felt…? and How did Anne feel…?). Technically and literally you are correct, since ‘est-ce que’ is itself the inversion of ‘c’est que', but in reality it has no meaning at all and has simply become a question marker.
3. Skipping the Pronoun
Essentially you can do this whenever, but the main driver is euphony. Here are the examples you pulled:
- De quelle région vient Eric?
- De quelle région Eric vient-il?
- De quelle région vient-il (Eric)?
The first one hits my ear wrong because the verb and subject are inverted so you expect to hear some kind of liaison but that combo doesn't sound right. Sometimes liaisons are not used with names to respect the nature of the name.
The second one is grammatically fine, but putting the name in the middle, it forces you to pause either side of it, in order to pronounce everything (mainly the name) properly.
I would be inclined to go for the 3rd option. It has a better flow, is more natural, leans on fixed phrases and I would simply specify Eric's name at the end only if I felt it necessary.
I hope I understand your questions correctly and answered them accordingly.