I strongly prefer 'there is an apple and a banana.' This sentence could be written as 'there is an apple and (there is) a banana' which I suspect is the reason why I prefer the version with verb conjugated in the singular form. As for your second question 'an apple and a banana' is the only correct possibility if you are talking about the individual fruits. It is possible to say 'an apple and banana...' but not on its own. If you were to eat something like a yoghurt which was apple and banana flavour, you could say 'I ate an apple and banana yoghurt this morning.' But if you're speaking about the individual fruits, both 'apple' and 'banana' need an article.

2) and 3) :)

2 and 3 sounds more natural :)

First of all, I'm not a native speaker, so well, sorry, too bad. But I think I can still help you with this one. Just remember that if you want to know what the right form of the verb is (or perhaps I should say the most likely to be correct form of the verb), you just need to care about the noun after it, if it's singular, then it's "is", if it's plural, then it's "are", you don't really have to care about what rest of the sentence. Therefore, 2 is more common, and keep in mind that 1 is not completely incorrect. Regarding 3 and 4, 3 sounds natural I think, quite frankly, I've never heard anything like 4, or perhaps I did but I don't remember.

Use "are" for this because normally "apple and banana" would use a plural verb. "There" refers to "apple and banana", so its "are" not "is". However, in practice I doubt anyone would even notice the mistake if you said "There is". I prefer "are", but I think both sound natural.