Question about game/giveaway

There's absolutely no problem with that :)

No, go right ahead. You're fine.

It's perfectly fine. If you don't have it in your own Steam library there's no problem in entering the respective GA's, even if you gave one away in the past.

You can enter for giveaways of game you gave away. You can also make giveaway(s) for game(s) you won. The only rule is that you're not allowed re-gift the key you won.

Some people enter giveaways for games at the same time as they give away their own copy...
Personally I wont ever enter for a game I gave away, but SG does allow it if you want.