(not sure how "2 questions" turned into 3 questions, but alright lol)
1. Timers and mats are provided by the delegate and/or organizers, but puzzles are supplied by the competitor
2. Whenever a puzzle is at the scrambling table, it will be associated with your score sheet, which will have your name on it. Whenever you're not competing and the puzzle is out of your view for some reason (it's common for people to let others try out their puzzles, but definitely not necessary), I usually just rely on knowing my puzzle. It's not super common for two people to have the exact same puzzle and get them confused, and whenever it does happen I can usually identify mine via the unique setup and potential scratches, etc. (Fun fact though: I accidentally traded Valk 2s at a competition, and theirs was imo a better setup, so I kinda lucked out on that)
3. I'd recommend bringing any puzzles you want, even if you aren't competing in that event (for example, even if there's no megaminx being held at the comp I'll still bring my megaminx, and I bring non-WCA puzzles), a timer (if you have one; it's nice to be able to practice with a stackmat before you compete), water, and maybe some snacks depending on the time/length of the competition.