The Official Team Fortress Wiki has names of some organizations that do league play. If you are in North America, the ESEA has an open division and pick up games for players of all skill levels.
These leagues may run the Standard competitive format, which is a 6 on 6 match with crits off and team composition limiting (only 2 scouts on a team at one time, etc.) and fixed weapon spread; Highlander, which is a 9 on 9 match with the same rules as SCF, but only 1 player per class per team; and fun competitions like Ultiduo or Basketball.
Due to the decrease in popularity in competitive play, the ESL shut down their TF2 ladders on May 5th, 2012.
From what I understand, these watermarks generally show up in Alpha/Beta/Early Access games.
Generally, they all have a similar format, consisting of some sort of unique identifying number; often along with the game name, the beta version, and your username.
There can be multiple uses for this.
Bugs/Information - The watermark can provide build information, and perhaps other information about what program version is running. This can make troubleshooting a lot easier when provided with a screenshot of an issue.
NDA/Privacy - Some games initially release under non-disclosure agreements and have very strict rules about sharing game information outside of approved channels. By including watermarks; you can mitigate the ability for people to leak content that you don't want released. It becomes easier to track down who released it (they might have more subtle ways too, but this lets the user know they can find you).
Based on the fact that most full-release games I've seen opt to not include this, I would think these are the two primary reasons.
I considered it as advertising the game and watermarking screenshots to be a third reason; but anyone who wants to steal your content could easily cut that part of the image out. I think that's a fairly unlikely motivation.
Best Answer
Yes and no.
The Arena modes, which are ranked and are required to qualify for tournaments, are fully cross-platform.
The online tournaments are separated by region and by platform, so mobile players play against mobile players, console players play against console players, PC players play against PC players, etc.
The major World Cup event in summer 2019 was played on PC. Most players used keyboard and mouse and some used controllers.