I think this might be an Australian slang. As someone from North America, I've never heard the term "flat out" used to mean busy. Unless you're planning to be around a lot of Australians, I would stick to using the normal usage for flat out. A: The sky is green B: That is flat out wrong! So in the example above it can replace "extremely", "clearly", "definitely" or it can be used to replace a word like "huge" when used as an adjective ("That is a flat out lie!).

I think you have to say I’ve been working flat out recently. If you leave out the verb, it might seem that you are flat, or lethargic. The fist sentence is clumsy, but understandable.

Yes it does

I don't think it would be the most appropriate word for your sentences. Flat out means: - At the maximum speed, level, rate, etc. that is possible: When you're working flat out, that's when relationships can suffer. - In a direct or simple way, without trying to make something sound more pleasant or more polite: She finally just asked him flat out what he wanted. - Completely or simply; used for emphasis: He just flat out lied to the American people.

The first sentence sounds odd to me (though I would get the meaning.) The second one is just fine.