I know that it says in the rules that the timer should end when I press "next" on level 10, but in the world record the timer stops when all the jelly is gone, so I was just wondering, since that could potentially save like 20 seconds, is that ok to do?

No

Isn't the record like 25 seconds off then?

Beating 6min 19 seconds will be pretty hard unless with good RNG, I'm going to submit my run soon.