hizandherz,
When I go to Tulum I like to rent a car and drive down. This will give you the ability to get out and explore the area. If you and your wife are more of "show up and stay at the hotel/resort" kind of people you can take a cab also. I like the freedom of having a car.
Ice cubes...depends on the hotel/restaurant. If they make ice out of tap water then no don't get ice. If they buy ice, that's usually from treated water and is ok. Some places also filter there water onsite which is fine as well. Just ask where ever you are, the source of the water for the ice. I "try" to never drink the local water. Out of the 10 times or so I have been to the Mayan Riviera including Cozumel I have gotten sick 3 times. (Once when I was brushing my teeth in the shower - like a fool!) I usually brush with bottled water but I forgot...I won't forget again!
There is not a shopping district per se in Tulum. But you can go into town and browse a few stores. Also, on the Hotel Zone road going south - there are a few local merchants that set up on the side of the road selling local crafts, nick naks, t-shirts etc... You will need a car to drive to these....it's also a nice drive along the coastal road. After you leave the main Tulum Hotel Zone there are a few hotels/condos further down the road.
Ants...I have never had a problem myself. But this is the tropics. There will be some bugs and perhaps ants...though having an ant problem in a room is probably not all that common. Don't leave food lying about your room!!
Have a great trip!