Can you use it like in English, in the context when somebody asks you "Do you want some tea"?
Yes, that is exactly the meaning.
Is the lack of a comma between "Ano" and "prosim" intentional, or is it a typo? Thanks
we picked what we thought worked better for the reverse translation. either way is allowed in czech, claims to the contrary notwithstanding:
Tyto vsuvky (prosím, celkem vzato, upřímně řečeno, bohužel, mimochodem apod.) si sice uchovávají větnou platnost, ale není třeba je od okolního textu oddělovat čárkami: Pošlete nám(,) prosím(,) všechny potřebné údaje. Je to(,) celkem vzato(,) věc nemilá. Nepůjdu tam(,) upřímně řečeno(,) ani za milion. Ve vašem případě(,) naštěstí(,) není kam spěchat. Ač je mi to velmi líto, zítra odpoledne(,) bohužel(,) přijít nemohu. Mimochodem(,) měli zavřeno.
Isnt "prosim" like a response to "thank you" like "dont mention it" also you guys have dobry den as hello, which is literally good day
Prosím can be used as both "please" and "you're welcome," and both "hello" and "good day" can be used for dobrý den. You will find that many Czech words and phrases have multiple acceptable uses, just as they do in English and other languages.
Quite Similar to «proszę» in Polish.