1. How does “уверь” function in this sentence:

literary style

2. What is “было” doing in this sentence:

“успел тоже защитить блестящую диссертацию о возникавшем было гражданском и ганзеатическом значении немецкого городка Ганау”
(было means it was about to happen, but didnt' happen; almost happened)

3. What case is “потери” in? Shouldn’t it be “после он терял кафедры”?

Потеря - noun here. = loss (i.e. потеря кафедры = loss of chair, loss of office)

4. Does this phrase sound normal?

“Очень могло это быть, потому что чего тогда не было?”

The “очень” sounds strange to me. Wouldn’t something like “действительно” or “непременно” work better? I wonder if this is a case of one of Dostoevsky’s narrators being a bit of a country bumpkin.

(not really country bumpkin, it's just informal, but quite normal (though it seems to contradict rules of formal syntax), очень is used for emphasis here and in similar phrases)

5. How does the phrase “и чуть не потрясшее здание” work in this sentence:

seems потрясти is used here in the meaning 'to shatter, shake' (to shatter a building)