You have asked some interesting questions. I'm just wondering what television channels you receive in Poland that give you the impression that the UK doesn't produce any, or hardly any, programmes?
My impression is that films are nearly all made in America, but that is to do with economics rather than language. The same is true of cartoons in particular, as films and animated cartoons are notoriously expensive to make well. However, there are still plenty of other programmes on every day which are made in Britain using British English. In fact I'd go as far as to say that most every day programmes are in British English. Presumably you receive a satellite channel of some sort?
To answer your questions:
1. No, I don't think British English is dying out. It is modifying over time, as it always has, by incorporating new words and phrases from different sources. However, American English also undergoes constant transformation by incorporating new words and phrases from other languages. When does something stop becoming the source language and become American English? We then might take such a phrase from American English into British English. So it's a complex and ongoing situation of incorporation and modification. It is important to remember that American and British English are just two great branches of one family, and of course the basis of American English is still English. Often they have retained older versions of words or phrases that went across with the settlers from Britain, whereas British English might have lost them since the 1600s. Theoretically, you may even have a case where one of those words is rediscovered by British English.
Ultimately, who knows, British English could die out, especially if one looks at the numbers of speakers, but as I have mentioned it is a complex and fluid situation.
2. In my experience the younger generation tends to use flavour-of-the-month words, which then drop out of fashion like the previous seasons clothes. Some things stick, others die out, and sometimes things which died out come back again. One of these was the word cool which first started being used in Britain in the 1960s and 1970s. It completely died out in the 1980s but then re-emerged again in the 1990s, and has since then gradually fallen away again.
One current phrase which is in fashion is to say "can I get a coffee?", meaning "may I have a coffee?". It will be interesting to see whether that usage sticks, or dies out.
We tend to absorb useful or "trendy" words, but tend not to use words which we already have, especially where those words are shorter than the American version. For example, British speakers will say that a house was "burgled" (broken into and possessions taken), whereas American speakers (I believe) will say that the house was "burglarized". Therefore, I can't see that one being adopted any time soon.
Another word which seems to be popular at the moment is "upcoming" instead of "forthcoming". Also a current phrase that seems to be used quite a bit is "heads-up", for which the British English equivalent is "forewarning". I can probably see those American English versions being retained as they are slightly shorter.
Many British people of all ages want to use British English rather than any other form, and so continue to do so. Others want to talk slang, but it should be remembered that British English has its own slang, and that also adopts useful slang words from American slang.
3. However, and this is the key thing; no-one in Britain adopts an American accent. They may say words or phrases adopted from American English, but they will say them with an English accent and intonation. For example, when someone from London says "can I get a coffee?" they don't say it like someone from New York, or California, or the Midwest, they say it with their own accent. I think the big difference is that we already speak English in Britain, and so haven't learned it from films and television, but from our parents and from each other. This will be different from an English learner's experience where they are exposed to a whole range of sources and accents, including those of other learners.
Student or Learner