Your knowledge:
1 - maybe
2 - yes. Any причастия.
3 - They have a краткая форма - yes
which is most of the times used.
- ??? Краткая форма is preferrable when and only when the participle is used as a predicate. The same rule as for adjectives.
4 - they have exactly the same syntax functions as adjectives. There are some spelling differences though, like no double "н" in adjectives.
5 - Transitive verbs (those, which have direct object) and only they have passive participles.

Structural and question:See here. Additionally:
1. Participles formed with -т- suffix are exceptions and to be memorized IMO. They are not so much, I believe.
2. Yes, sometimes stress is moved to suffix then е is turned to ё. When it is moved and when is not sometimes is a problem even for native speakers. Sometimes both ways are acceptable.

Look, human languages is not a mathematics. I believe you can easily find many illogical phenomena in Greek as well if care to search.