This is not the way to approach getting a job as a game designer. It won't happen, not this way. At least not with any company worth working for.
First of all, if you have to ask what a professional game design document looks like, even though you say you can make it "look professional", you are not getting the job. Game designers have worked on several projects, they have a proven track record, and they certainly know the formatting of a game design doc.
I know this all sounds harsh, but it's the reality.
Be a game designer
You can be a game designer without working professionally as one. There’s nothing stopping you from designing games. And if you want to do this professionally, that’s exactly what you have to do. You should have many finished game designs, not just on paper, you should make your own working prototypes.
As you likely won’t have anyone to make your games for you, you need to do them yourself. We’re not necessarily talking finished products you can release on Steam, but it would help a lot. If you can prove you’re able to come up with interesting gameplay mechanics all wrapped up in a nice package that actually sells, well, then you’ve got a lot better chance than the guy who just sends in a game design doc… that document will likely not even be read.
The game designer position
I don’t think the game design position is what you think it is.
There are very few game designers in the world doing just game design. Most have a lot of other responsibilities. If you do work solely as a game designer, well, then your job on that project can be over before you know it. It’s very likely you won’t get to be involved in a project from start to finish, you might be there just from the start and do a game design doc, and then you’re done. From there someone else is in charge, like the producer.
Game design is more about game rules and mechanics than it’s about “this MMO where we have the whole planet in our game, and you can just like, live in apartments, buy things, and yeah, there should be discos where you can be like all virtual reality on the dance floor, when you’re actually at home in your apartment, but in the game! So it’s a game, but VR, inside of a game. And, oh yeah, there should be…”
Getting the job
Basically I’m saying you can’t get a job as a game designer without having designed a few games. It’s as basic as that.
There are a few ways to get there. A) like I’ve said earlier, you can start making your own successful games. B) You can work on countless games in other roles, testing, programming, concept artist, whatever it takes, and eventually get to the producer role. From there you can eventually get to a game design position.
The game design doc
To actually answer concretely at least one of your questions: