Jacket is short while a coat can be either long or short. In English the definition of coat has become muddled. An overcoat has or is becoming archaic so "Your coat is completely wet" is correct
is it the same as Russian куртка and плащ? I would translate куртка to coat, but here they don't accept "coat" for "kurtka" (you have to put "jacket"), though I don't really see a difference between coat and jacket
In my opinion yes, it's the same, these are clearly same words, just in different languages of the same family. Of course there are false friends sometimes, but not here.
I would also translate kurtka to coat, but it seems that we both would be wrong - when searching for 'coat' in Google Graphics, almost all results seem to be what I would call 'płaszcz', and 'jacket' shows what I know as 'kurtka'. So I guess they're right not to accept 'coat' for kurtka.
I don't think that I would ever use плащ in Russian though, I just call evertything куртка (maybe that is not exactly correct)
I'd translate coat as пальто, which has a somewhat different meaning from плащ. As I use these words, they are from different materials and for different weather: плащ is for rain, пальто is for winter (and шуба is for very cold winter)
Well, for sure for "cały" you need the noun to be masculine, because this is an adjective and it has to match. Generally it's accepted in some collocations, but I'm too tired to think of any others than this one. "zupełnie" simply means "completely".
From what I was told by a native, this seems quite strange in this sentence.