"I do not want to talk with him."
Translation:Nie chcę z nim rozmawiać.
As a native speaker of Polish, I would say that both are correct, but have a bit different meaning.
- Nie chcę z nim rozmawiać - I do not want to talk with him (at this moment).
- Nie chcę z nim mówić - I do not want to talk with him (never ever).
I cannot say if these niuances could also exist in an English phrase.
Actually, Polish Wiktionary puts "rozmawiać" as one of the meanings of the word "mówić", with the example: "Chciałbym mówić z prezesem." Słownik Języka Polskiego - doesn't. It does not seem to me that it is used often in such a way, and "Nie chcę z nim mówić" seems quite weird to my ear, even if it's not technically incorrect.
"Nie chcę z nim mówić" sounds very weird to me. "Mówić" is more like speak or say, one can "mówić DO kogoś" rather than "z kimś", unless they are speaking at the same time. Even saying "I won't never ever talk/speak with/to you again" would be " Nigdy więcej nie chcę z tobą rozmawiać!" not mówić.
Actually, that is the reason I thought it should be correct, because in my native language, namely Croatian, which is a close relative to Polish, there is the same dichotomy between "govoriti" - to speak, and "razgovarati" - to talk (to sb), and we can also say "govoriti s nekim" - to speak with sb, which sounds quite formal.
Wow this article not only have me some insight in the perfective verbs, but also a whole new view on all verbs; very good. One thing that I noticed, though, is that you often translated perfective verbs as some action that happened in the past. Is that always the case? Can we assume that a perfective verb refers to something that happened (or at least started) in the past? Thank you
Perfective verbs have forms in past and in future, but not in present.
Imperfective verbs have forms in past, present and future.
You can check all the forms of any Polish word using Grammatical Dictionary of Polish http://sgjp.pl/ (well, maybe not all the words, but this dictionary contains a huge amount of words that are not listed by any other Polish dictionary available online). The only problem is that it does not explain the meaning of words, unless it is needed for disambiguation.
E.g. check "rozmawiać" http://sgjp.pl/leksemy/#252406/rozmawia%C4%87 - you have there
- Indicative mood
- Imperative mood
- Conditional mood + other forms
Check "porozmawiać" http://sgjp.pl/leksemy/#252408/porozmawia%C4%87 :
- Indicative mood
- Imperative mood
- Conditional mood + other forms
However, I need to mention, that in past tense Poles often use imperfective verbs in reference to completed actions that lasted a longer time (would this be half an hour or half a century).
... when I move to Poland.
If you plan moving to Poland, I might suggest you watching some episodes of "Love My Poland!" channel https://www.youtube.com/channel/UCrdgjF4pP_xvQMXmKvAS7uw - Russell is an American living in Poland and he brilliantly speaks about the cultural differences between Poland and America.
The usage, as you mentioned, of imperfective for past, completed actions is it grammatically correct?
I was thinking of that... and in my opinion that matter is outside of grammatical rules. This is rather a matter of style.
That is sometimes hard to explain, but I will try with some samples that - excuse me - may be not very accurate, because it is really a blurred field of the language usage and depends a lot on the context, intonation and the exact verb used, especially when speaking about the past. In some contexts and with some verbs, there is really no big deal whether the perfective or imperfective one is used. Especially when speaking about the future, the choice of perfective or imperfective verb informs rather whether we expect the action to be completed at a time (perf.), or rather an activity is expected to last for a longer period of time or will need to be repeated several times (imp.).
I hope, that other persons also add their comments on that.
When used to describe the past, imperfective verbs are used in "softer" language, while the perfective ones are used in more "incisive" language.
- Wczoraj rozmawiałem (imp.) z moją mamą. - I talked with my mom yesterday (she was happy to hear some news from us finally)
- Wczoraj porozmawiałem (perf.) z moją mamą. - I talked to my mom yesterday (I had to tell her finally to stop sticking her nose in where it's not wanted).
- Pisałem (imp.) notatki podczas tego wykładu. - I wrote notes during that lecture (it is good that I have them).
- Spisałem (perf.) notatki podczas tego wykładu. - I wrote notes during that lecture (and I am going to use them some time!)
- On cały dzień leżał (imp.) w łóżku. - He stayed in bed all the day (poor boy, he must have been very ill).
- On cały dzień przeleżał (perf.) w łóżku. - He stayed in bed all the day (that is outrageous!)
- Prosimy (imp.) o dwie kawy - Two coffees, please (fairly natural request in a café).
- Poprosimy (perf. future!) o dwie kawy - Two coffees, please - the future form of the perfective verb used in the meaning of present tense request - for the absence of its present form - has a stronger shade of... politeness; see also the comment at the bottom.
Imperative with negation: we normally use imperfective verbs. Some of the perfective verbs (but not all of them) may be used for specially strong statements, but then extra words are usually added to put even more emphasis on the phrase.
- Nie zamykaj (imp.) drzwi. - Do not close the door (it needs to be open).
- Nie zamknij (perf.) drzwi przypadkiem. - Do not close the door by chance (it is really important / do not even dare to do that).
- Nie rozmawiaj (imp.) z nim. - Do not talk to him (it is of no use / I do not want you to do that).
- Nie porozmawiaj (perf.) z nim tylko. - Do not talk to him by no means (I strictly forbid you do do that).
Imperative without negation - we may use both prefective and imperfective verbs, but it is just opposite than in the indicative mood - perfective verbs are "natural" or "neutral", while imperfective are "stronger" and more "agressive".
- Zrób (perf.) teraz obiad. - Go and make dinner now (it's time already). 
- Proszę zrób (perf.) teraz obiad. - Please, go and make dinner now (it is high time already / we really need it now). 
- Zrób (perf.) może teraz obiad. or Może zrób (perf.) teraz obiad. - Go and make dinner now, maybe (using "może" = "maybe/perhaps" instead of "proszę" makes the request even softer, more like a suggestion, and thus more polite).
- Rób (imp.) teraz obiad. - Go and do make the dinner now (how could you be making something else!?)
- Zapisz (perf.) to co teraz będę mówić. - Write down what I am going to say now (it is important).
- Pisz (imp.) to co teraz będę mówić. - Write down what I am going to say now (are you even listening to me?)
- Porozmawiaj (perf.) z nim. - Talk to him (it is my suggestion, because I think it is a good idea).
- Rozmawiaj (imp.) z nim. - Talk to him (it is important! / you just have to do that).
 Skipping "please" when speaking to close persons is very natural in Polish, however rude might that seem in English. Even more than that, it is rarely used when talking to strangers, too, and in some circumstances adding "proszę" at the end of a phrase might be even frowned upon - it sounds as extra insisting. That is sometimes a reason why Poles may be considered rude in USA or England for skipping "please", while we simply use the code of conduct of our language and we tend to assume that others are willing to cooperate, but do not want to be begged for something. That is why you'd rather hear a Pole speaking English to use subjunctive (like "would that be possible that you pass me the salt"), than to add "please" at the end of a simple request...
 "Proszę" at the beginning of the phrase is more polite - still insisting, but less than when "proszę" is at the end.