Right. It's a singular imperative, so if you use «ты» with a person, use this form. But if you address someone with «Вы», you should use a different form (поговори́те).
Ok. Shoudn't we have imperfective since the meaning is more likely to be "join me in a conversation" than "have this talk with me"?
Affirmative Imperative forms of imperfective verbs are used (1) for impatient requests ("Why aren't you talking to me? Do talk to me") and (2) orders ("Talk to me. I order you"), whereas the perfective imperative forms are used for less emotional polite requests ("Will you please talk to me" = Поговори со мной, пожалуйста). Only imperfective verbs are used in negative requests ("Не говори(те) никому!" = "Don't tell anyone!"). The negative imperative of perfective verbs is synonymous with the if-clause of the unreal conditional: "Не поговори он (=Если бы он не поговорил) со мной вчера, я бы так и не узнал, что произошло" means "Should he not have talked / If he hadn't talked to me yesterday, I would have never learned what had happened". There are also a few cases where negative imperative forms of perfective verbs are used idiomatically, e.g. "Не скажи(те)!" means "It isn't necessarily the case!"/"I'd rather disagree with you". Other examples include biblical commendments (Не убей, Не укради, Не сотвори себе кумира), although in more modern translation of the Bible imperfective verbs are used instead.
This isn't confirmed but I'm pretty sure you use it when the following word starts with two consonants
We use the preposition со istead of с, when the following word starts with two consonants of which the first one is в, з, с, ж, ш, or м. Exеptions to this rule include phrases with the words враг =enemy (с врагом, с врагами), млекопитающее = mammal and Млечный Путь = Milky Way (с млекопитающими, с Млечного Пути).
Likewise, we replace в, к, из, над, под, перед and от with во, ко, надо, подо, передо and ото, respectively, before certain words starting with two consonants. Here are some examples: во благо, во время, во времени, во введении, во всех, во всём, во всей,во главе, во главу угла, во двор, во дворе, во зло, во зле, во избежание, во лбу, во льду, во мне, во мгле, во мне, во мраке, во рту, во рву, во славу, во сне; ко времени, ко всему, ко всей, ко всем, ко двору, ко лбу, ко льду, ко мне, ко рту; изо всех сил (with all one's might), изо льда (из льда is also used), изо рта;от/ото всех, ото лба, ото рта, ото сна; надо мной, подо мной, передо мной, над/надо всем/всей/всеми, под/подо всем/всей/всеми, подо льдом.
No! "Поговори мне!" is a threat, something like "Just you dare to open your mouth, you nasty little thing!"
So, I'm guessing this could be use in such a case as: My sister is angry at me and I tell her to speak to me, tell me why she is angry
Something taumatic happened and I'm telling X person "Speak with me!" as in a, c'mon take it out of your system, kind of way?
You're right. It could also be used when your friend's or close relative's talk makes you feel better. There is song that was popular in Russia in 1970s which starts with the phrase "Поговори со мною, мама". Check it up on YouTube.