Я стал занят - I've become/became busy.

Мови - It's a Ukrainian word meaning "language(s)" as far as I can guess.

А если хочешь - hmm, you can just get rid of "a" there, no real change in meaning. А is often used to introduce a question and I think "а" qualifies as an interjection in such cases.

А ты не знаешь, что завтра показывают?
А что ты делаешь?

And blah blah blah

Actually, come to think of it, maybe "а" there has a similar meaning to "and". For example, something as mundane as asking:

Как у тебя дела? vs. А как у тебя дела?

The first one is most likely to be used to start a conversation with someone. The second is more likely to be used when you've already been talking to someone for a bit but never asked how they were doing untill the moment. In that case:

А как у тебя дела? would very roughly mean something like:

In addition to everything that's already been said I'd also like to know how you are doing

But it's the grey area, no doubt of that. Explaining "а" is like expaining the difference between sentences like:

Ну ты молодец! - Ну ты и молодец!

I hope that this messed up "clarification" will do any good