"We always study together."

Yes, but the correlatives have their own endings that are unrelated to the usual -e -i -o -a.

Similarly, "kiu" is not an imperative, for example.

There are also other adverbs that don't end in "e", such as "ĵus, nur, nun".

you are not the object being influenced by the verb. instead, in this case, you are the one doing the action. a good example sentance of the accusative is "Mi manĝas la pomon" You are the topic and you perform the verb upon the said object (i.e. the apple). Hope this helps

Because that's the infinitive, but you need the present tense here (for something that's always true).

It would be as if you said, "We be smart" (with infinitive) rather than "We are smart" (with present tense).