For “complete 20 crucible bounties”, Can I finish all the bounties and just leave them completed in my inventory to turn in once I hit that step or do I have to complete them fresh
Yes. You can. But once you get to 15 stacked up, Shaxx won't have many that you don't have, being Im pretty sure there is also 20 total bounties for the crucible in the game it will be hard to stack up all 20. Also if you stack up say 17, then you get to that step and turn them in, and it is after that days daily reset (1:00 EST) you will have to wait until the next one as it will give you the "You have already completed this" message.