I'll second some of the responses here and say that unless you're gifted, you probably won't be able to avoid confusing Kyokushin with Taekwondo. The stances, the techniques, the level of force, what counts as "legal" or "illegal" contact, etc. will all be different enough that it will drive your instructors crazy trying to correct you all the time. You're going to find yourself trying some technique or strategy from the other martial art you're training in, and your instructor will tell you to cut it out.
In some cases, instructors throw students out for training in other martial arts at the same time. I don't agree with that. I think it's silly to essentially ban a student for just trying other martial arts at the same time. The best instructors I've ever encountered have actually encouraged students to go train elsewhere. But that being said, it happens. So you need to ask your prospective instructors how they feel about it before you sign up. Otherwise you might get in trouble later on.
What I recommend is to try a small number of classes in one martial art, then a small number of classes in another martial art. A lot of places give you the first week of classes for free. Or at least you'll get one free introductory class. Then just pick the martial art that you liked the most, and spend all the time you have available on that one martial art. Get good at it. Maybe stay until black belt, which is typically a 4 year endeavor. After that, try other martial arts.
As for advantages to doing multiple martial arts at the same time, it really depends on which martial arts we're talking about. Taking martial arts that are very dissimilar has huge advantages. For example, taking Boxing and Gracie Jiujitsu. Then your brain doesn't get confused, since the techniques and scenarios are so different. And so you can certainly benefit from training in both at the same time, since they cover very different topics.
Also, when you get really good at one martial art, you can add a second martial art fairly easily, without the confusion you would get if you started taking them both at the same time. And this applies to even similar martial arts such as Kyokushin and Taekwondo, or Wrestling and Gracie Jiujitsu. It's because you have gotten so advanced that even the small differences between the two styles seem quite large in your mind. You can instantly see it. It's not confusing.
As for your third question, about tournaments. Yes, let's say you are actively training in both Taekwondo and Kyokushin. You go to a Taekwondo tournament, and you say you're from So-N-So's Taekwondo. No problem. You go to a Kyokushin tournament, and you say you're from Who-Z-Whatsis Kyokushin. No problem. But if you enter an "open" tournament (open to all styles of martial arts), then you have to tell them you're from one, but not both. So register as either Taekwondo or Kyokushin. It doesn't really matter.
I've never seen a tournament that has both a Taekwondo division and a Kyokushin division. But if those exist, then you might be able to register for both divisions at the same time. The only problem you might encounter is if both your Taekwondo and your Kyokushin events are scheduled at the same time. In other words, if there's a time schedule conflict. So you have to pick one or the other at that point. Oh, and you might have to perform a quick change from one gi and belt to another gi and belt. That's because each school has its own uniform, patches, insignias, and belt colors.
One other thing about having rank in multiple styles while entering a tournament: Let's say you have a black belt in Kyokushin karate, and you just started to take Taekwondo. So you have a white belt in Taekwondo. If you then enter a Taekwondo tournament, you will win at sparring in the white belt division, obviously. It's not fair to others. And their instructors will try to have you disqualified, because you're obviously not a novice.
So you might compete in the black belt division for sparring in that case, even though you only have a white belt in Taekwondo. The way you do that is by explaining yourself to your instructor. He or she will then get the tournament officials to agree to it. Your instructor will award you a temporary black belt to wear just for that sparring event. And then he or she will remove it after you're done.
Well it sounds like you're just beginning to figure all of this out. Good for you. I hope you find what you're looking for. Good luck!