When is the best time to put in a flight packet? Does rank matter?

Rank is immaterial to the board and branch, however accessions can be difficult due to shear number of techs for a aviation packet. My first packet went in at PFC, then SGT and finally SSG before selection. Start now snd never give up if it is something you wish to do!

Rank does not matter but you have to build a resume for your packet, so will be smart if you give it a shot at your job first and then apply, you have 40% more probability of been selected as a SPC than you have now

You've received much good advice here that revolves around one central theme; apply, early and often. At least one response was that your résumé matters and that you should build that before you apply. I spent many years doing flight school interviews, reviewing candidates packets, and helping them polish them. I have at latest count, done so for 22 successful selectees, the first of which has been a CW4 for a few years now. Even with all that experience, I gave up on trying to divine what "the board looks for". That seems to be a much moving target and difficult to pin down but one thing remains true; your packet must be complete, error free, and professional. That includes any letters of recommendation. If one of your recommendations is full of typos, spelling and grammar errors it reflects upon you as well. Don't think that matters? You're right. It doesn't matter to 75% of the people who grade your packet but the other 25% also have a vote and they notice everything.

Get as many people as you can to review and critique your packet. The guy who pulls out a red pen and AR 25-50 is the guy you want. If a LOR is full of errors either pull it and get another or take it back to the recommender with the corrections already made for their new signature.

When it comes to resume building, again don't worry about the board wants. Build it with a view on three things:
1. Good for promotion points.
2. Good for a civilian career.
3. Something I want to do for me.
After all, there is life after the military; injuries and illness curtail the career of many a great Soldier. Be prepared to not be selected for WOFT, to not be selected for promotion, to not be able to continue to serve. If you find things that meet all three criteria, do it.

All in all, good luck to you and keep after it. Getting selected is the hardest and the easiest part of becoming an Aviation Warrant Officer.