"Beat the game" is probably the most popular way to say that you completed a game. However, it does not necessarily imply that you have completed "everything possible", like OP is asking. It's more likely to imply that you completed enough to see the "ending", which is especially misleading if the game has multiple endings, and even more misleading when there is only one true ending, per se.
It means that every level was completed if the game has to be played in a linear/sequential manner (eg if every level must be completed in order to progress complete the game).
To be explicit (which is what OP seems to want) that you completed every level or mission, you should include something, like
I beat every level.
Actually, the example in your title seems a little formal, but fine nonetheless.
- I have completed all the missions/levels in the game.
If you and the listener understand that you are talking about a game, then "in the game" seems redundant.
- Finished it all?
- Have you finished it?
- Have you finished it all?
These all seem fine, but 2. sounds the most casual.
- Have you played it all?
- Have you played the whole game?
These questions sound fine, but they do not necessarily ask whether or not the listener has completed every level/mission. As you know, in some games, you can play the whole game, but not complete every single mission/level. (I cannot think of any at the moment.)
Some alternatives are "pass", "beat", "complete":
- I beat/passed/completed every level/mission!
- I beat/passed/completed all the levels/missions!
- I beat/completed the whole game!
- Have you beaten/passed/completed every level/mission?
- Have you beaten/passed/completed all the levels/missions?
- Have you beaten/completed the whole game?
I think "I passed the whole game" and "Have you passed the whole game?" might sound strange. This list is not meant to be exhaustive and I'm sure there's plenty of slang out there that will work.