Personal opinion: I don't like Sentence 1a; it seems to read awkwardly. Something about the don't only throws me. Here's how I'd rewrite it:
I don't just study a lot; I also play a lot.
Sentence 2a is okay, but I might be tempted to repeat the subject (this is not a required change, but some might feel it helps the readability):
I not only study a lot but I also play a lot.
Because it starts with not only, Sentence 3a needs the verb do. I would also add a comma and repeat the subject:
Not only do I study a lot, but I also play a lot.
As far as the negations go, I think Sentence 1b is better than Sentence 1a, but I'd still reword it like this:
Not only do I not study a lot, I also don't play a lot.
You don't really need the "but" in that sentence. The "not only" alerts the reader that a contrast is coming.
Some of these changes are more about improving the sentence as opposed to correcting grammatical errors. In other words, not only do they improve readability, they also improve the sentence's natural flow.