Hey! It's awesome to hear that you love the game so much and that you want to get into the competitive scene.Let me try and answer all your questions 
Firstly, for PC, I would highly recommend learning to play on a mouse and keyboard. The speed and precision that you can gain from using that can make all the difference in a skilled engagement.
We currently only offer the Pro League on PC. We used to have a Pro League on Xbox, however have decided to focus our efforts on PC for the moment.
There are still smaller competitions held on Xbox (via ESL) as part of the "Go4" series, but you'd need an Xbox to play in those.
Your Ubisoft account can be linked to your Xbox account, to share Ubisoft Club rewards, levels, and points. It doesn't allow cross play between the platforms though, nor does it save your in-game progress. Basically, you'd need the game for Xbox, and you'd need the game for PC. Your PC progress would not save on your Xbox (and vice versa).
There are tons of teams always looking to pick up new players. I recommend checking out Reddit pages like this one and engaging with the community to find new players to team up with.
We also have a reddit page where we talk about nothing but Rainbow Six Esports. You might like that too
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