Hey killo
welcome! and I'm glad that you're interested.
1. I'm sure the native speakers can elaborate more on this, but I've been studying the language 2-3 years, and am actually right now writing a paper on an aspect of the language, so I was reading a bit about this stuff earlier today
One of the main differences between the dialects is that the one in Montenegro is called 'ijekavian' and the one in Serbia is called 'ekavian'. This is minor, but noticeable in some verb infinitives and nouns.
the article i was reading today was about Serbian language before Montenegro declared independence, so it has some interesting comments. I can send you the name of the article if you're interested
I am not sure if in Montenegro there is the same propensity like in standard Serbian to construct verbal sentences like 'present tense + DA + present tense verb' (as opposed to how in Croatia and Bosnia, it's common to create a verbal sentence like 'present tense verb + verbal infinitive".
2. From what I've heard from my friends in Serbia (I would assume it's the same in Montenegro), one would learn both alphabets in school. depending on the schools, I've heard that one year students do things in Cyrillic, then one year in Latin. However, I think that both are acceptable unless otherwise specified if you're referring to assignments and the like. I have no idea but would be interested in what it's like, if there is any resurgence of one alphabet or another in schools now. Cyrillic is used a lot on official documents and the like, and street signs, etc.
I hope this helps!