Quetion about steam, gifts and steam inventory

1.- If it's in your inventory, it's a gift, so yes, you can create a giveaway for it.
2.- For those sites you get keys, so no, you can't have it on your inventory, as keys redeem the game automatically to your library. When you choose "send as gift" you will be sending a link so they can get the key directly.

1) Check that the gift has not been sent already. When you click on it, if you've sent it previously, it will say so.
The gift will not leave your inventory unless the recipient claims it (or unless it was sent directly on Steam, in which case I believe it leaves inventory immediately).
2) No game you buy outside Steam can be in your inventory as they will be given to you as keys.

1. gdy aktywowałeś Frozen Synapse Prime na swoim koncie, dostałeś w prezencie 2 kopię, też tak miałem. Z tego co pamiętam ta kopia ma chyba blokadę na region, w którym została aktywowana, więc dla pewności wybierz region Polska przy tworzeniu giveawaya

You got Frozen Synapse Prime because it was a two-pack deal at one time or another, so when you bought one copy, you got a free gift copy to give to a friend. If it's in your inventory, it's 100% unused and you can send it to whoever you like (even as a giveaway here).

Keys will NOT give you gift copies --if you activate keys on your account, it will add the game to your library (unless you already own the game).