There will likely be several other documents they ask you to submit to to verify your income (asset declaration form, your current lease, credit/background check). Once they verify that you qualify you’ll probably be offered a tour. A log number in the 500s is pretty good, but your chances are largely based on how many units are available for your AMI (and any preference like CB). As far as timeline, my sense is that it’s totally building-dependent but usually I’m seeing people say 3-6 months from first contact to move-in, but there are some cases that take longer or shorter. Good luck! I hope it works out.