I think what they're telling you is that you'll need a bag to get all the issue pubs from the training center to your hotel room on either the first or 2nd day. A small suitcase or medium-large backpack or two helmet bags or really any old bag of similar size will do fine, and you'll have ample time while you're going through ground school to buy a flight bag to use on the line. Unless things have changed since the January class, there's no need to arrive with your flight bag already in hand.
Don't know if they'll payroll deduct the flight bag or not.
As for fine footwear, black Ropers are ideal, and the Justin variety without metal is very popular. If you haven't bought yet, consider getting the higher rise ropers -- prevents pant cuffs from catching on the top of the boot when you get out of the seat. That with the rubber non-slip sole is probably what Sporty's offers, and lots of guys at SWA swear by their Ropers.
You probably know this, but "cowboy boots" with pointed toes are not permitted, but the rounded-toe Ropers are specifically allowed, and are common.
Congrats on starting class -- it's great to see you guys coming aboard!
Snoopy