Learn English – Phrase “sporting person”, correct usage

Is it gramatically correct? I'm not sure. A computer is not something that can be played. You can play on the computer, and you can play a computer game.

Is it acceptable to say? Absolutely! I say "playing computer" all the time. Actually, I probably say this more often than "playing computer games". Everybody will understand your meaning, and I doubt that anybody would think it sounds strange.

Now, StoneyB has a good point, if you say "playing the computer" (or even "a computer") it implies you are playing against a computer. This would sound a little bit strange unless someone is asking who you are playing against.

What are you doing?

"I'm playing computer." OR "I'm playing on the computer." OR "I'm playing computer games."

These all sound perfectly fine.

Who are you playing against?

"I'm playing the computer." OR "I'm playing against the computer."

Both of these also sound fine.

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.