Can I purchase something from the museum store without having to purchase a ticket? (Was there previously, but didnt have time to look around)

Yes i think so. Because the gift store is where you purchase your entry. Although you are on otherside of gate. Talk to someone there.

That should work. Their ticketing is in the gift shop.

You can go straight to the store and buy whatever you want without purchasing an entrance ticket.

Yes you can however you need to tell the attendant.