"We will talk after the lesson."
Translation:Мы поговорим после урока.
Yes, we use it in most cases.
It's kind of like, "We we will talk after the lesson."
Not really. English also has a -s ending in 3 person singular, but it doesn't mean you can drop the pronoun in "he goes".
Not always, usually it's absolutely normal to say "пойду" instead of "я пойду", "сделаем" instead of "мы сделаем" etc.
As I understand it, adding the -ива- or -ыва- infix makes a perfective verb imperfective. A present tense verb can be used colloquially for future actions (e.g., I am doing it tomorrow), but the true future would be either the imperfective future "будем ...-вать" or the perfective future "поразговорим" (or simpler поговорим).
Разговорить is something completely different, 'to get someone to talk'. Разговорить is usually used with -ся, разговори́ться 'to get oneself to talk, to start talking'. It's a perfective verb, so you don't usually add по- to it.
We normally don't add по- to perfective verbs. But if we did, then поразговорить would perhaps mean the same thing as разговорить, but would sound old-fashioned (e.g. old texts might use покупить instead of купить, поиспытать instead of испытать — the latter sounds as if taken from some bylina, like поиспытать силушку богатырскую...). I wouldn't use поразговорить unless I were writing a story about bogatyri getting their enemies to talk, or something like that.
Разговаривать used to be an imperfective from разговорить, and it can be sometimes used in this sense (I could find only one example in Google: «"разговаривать" пленных умели обе стороны» 'both sides knew how to get prisoners-of-war to talk', from a forum discussion), but this meaning is very rare. It has undergone a meaning shift: now it usually means just 'to talk'. Basically, разгова́ривать is a synonym of говори́ть now.
(I think that «разговориваться» 'to be getting oneself to talk, to be starting to talk [several times]' lost -ся at some point of history, and «разгова́ривать» changed the meaning to 'to be starting to talk [several times]' → 'to talk'. I'm not sure if this is historically accurate, but this explanation seems reasonable.)
Поразговаривать is basically по- + разговаривать, that is, 'to talk [for some time]'. По- makes a perfective out of an imperfective, and it basically means the same thing as поговори́ть (although поговори́ть is much more popular). I think it should be accepted too.