No, I don't think it's necessary to hold the advanced tuck position for the full minute before progression. I originally did work up to the full minute, but that seemed to have little carryover to the next step. I then went to the 1/2 tuck lever (one leg extended straight, the other tucked) before working the straddle.
The 1/2 tuck will work well with your low ceilings. Start by lowering in the advanced tuck, then extend one leg out as you lower and gain the clearance. For additional fun, once you gain some strength in this position, try "bicycling" your legs in the 1/2 lever position.
One thing that I have found, and this applies to all of the strength positions, is that you need to maintain a significant volume of work in the holds. If you don't, you'll end up regressing. What I mean is that if you are doing 60 seconds of holds in the advanced tuck, then you move to the half tucks, you'll likely drop to less total hold time as the intensity creeps up. You'll progress for a while, but then stall and then maybe even regress. Finish with some work in the easier versions to up the total hold time to avoid this.
Chest activation will greatly help your strength in the front lever. Besides the static holds, be sure to work some negatives in as well. As we are looking for elite strength, not competitive skills, the eventual goal shouldn't really be the front lever hold anyway. Full range strength with straight arms from hang, thru front lever to inverted hang should be the end goal. Negatives will help work toward this.