I agree with the above. Apply as soon as possible to get your stuff in as you never know when the windows will close. Currently the Fedex window is closed to new applications. As far as the physical, get it to apply then I wouldn't worry about getting it renewed until you get an interview.
As far what qualification looks like to whom, don't worry about appearances. You have to qualify for all then dance with who brung ya. You can always explain that "brand X" is your first choice and you were trying to make sure you qualified for all in case they (brand X) didn't call.
My $0.02.
JC Dude