I play on mobile and it’s pretty common for me to have to relog after going into the nether due to chunks not spawning. As long as the recording remains continuous, am I allowed to relog or does that DQ the run?

If the timer stays running, I'd assume yes, although I don't know for certain.

I'd say yes as well, as the timer will keep running when your outside the game but we have allowed runs that have to relog in the past so you should be fine.