Let's look at Present Simple. As far as I understand there is [almost] no difference between
I play this game. I do play this game.
The first question: can I form negative sentences this way?
(1) I play not this game. I do not play this game.
I know, (1) seems weird, but I have found a lot of examples of using this:
Forget not the tyranny of this wall, ... 
Death or healing, I care not which you seek. 
Speak not those words. 
The use of (1) seems quite logical to me as for beginner English learner. So, if we look at Present Perfect:
I have played this game. I have not played this game.
We can see the same (or not?) way of forming negative sentences. Present Continuous:
I am writing now. I am not writing now.
It's still logical for me. And now Past Simple:
I played this game. I did play this game.
The second question: can I form negative sentences this way?
(2) I played not this game. I did not play this game.
(2) is VERY weird, but seems logical. Or not :)