1) Do all boats leave from the same area?
No - there are several marinas, and dive ops all along the west side of the island. There are some that leave from Palm Beach, all the way down to Baby Beach. That said, depending on where you will be diving on any given day, most ops will either pick you up on the beach, or by car/van at your hotel and take you to the marina where the boat is.
2) Are there better dive ops to go with if you stay on Palm Beach?
Can't compare better or not, as I have not used Dive Aruba. The only op on the island I have used is Native Divers - Romeo is a lot of fun and I really enjoyed diving with him. While his boat is not small, he only takes small groups. I think 6 is the max he will take. He likes to cut up and will let you dive to your experience and profile. Again, most of the ops will pick you up by boat or van - even JADS which is at the complete other end of the island.
3) Are there any dive ops that will store your gear?
Yes, but I am not sure which ones do and which don't. I know the hotel dive op where I stay had lockers available... but I am not a fan of cattle boats, so I haven't used them and can't help w/ any details.
4) Do any dive ops dive on Sunday?
Yes, though I can't say for sure that all of them do.
5) Non-diving question: Does Aruba have good discount shopping like St. Martin, etc.?
Shopping is plentiful, and while prices on some things is good/fair - I don't think you can call it "discount". There is are a lot of shops and stores in Palm Beach or you can go to Oranjested for a bigger selection - a short cab ride (or the bus fair is really cheap).
Wacky Wahoo is a great place to eat - you may need reservations this time of year.. It is close to the hotel district - an easy walk, and the locals will point you in the right direction. We have done the sunset cruise/catameran as well, was a great way to spend an evening unwinding.
Have a Balashi for me.