I only write down the moves if it's 90+30 where it's mandatory. For me, writing down the moves breaks my concentration because I'm constantly thinking about writing the notation correctly instead of the actual game. It also takes up a lot of time if you're playing a short time control. Others might have a different experience though.
Do you record moves for otb rapid games?
10|0 is blitz.
Standard rapid is 15|10.
Yes, I record moves over the board, though it is not mandatory for rapid.
This is a tricky question. While many coaches here in Malaysia force their students to record 20-minute games, I personally find it a waste of precious thinking time.
Even though these are rapid games, it is still useful to analyse them. During the breaks between rounds, enter the games in a lichess study. Most players above 1500 can generally recall about 80-95% of the game, which is more than enough material to analyse.
Writing down moves maybe takes enough time to decrease playing strength at 10 minute speed. Longer games seem more worth recording. The short ones are full of blunders by comparison.