The Caye Caulker Water Taxi boats and other water taxis will take you to Caulker from San Pedro or from Belize City.
As to places to stay, these are some of the places I recommend. Most of these places are in the moderate range -- US$30 to $60 (which would be the budget range on Ambergris):
Chocolate's -- just room, and it's a beauty, plus Chocolate is a nice fellow
Tree Tops -- clean as a pin rooms, and the "guard dogs" here (little terriers) are an extra plus
Anchorage -- nice place at the south end of the island
Shirley's -- another nice placea the south end of the island
Seaside Cabanas -- new thatch cabanas just to your left as you get off the water taxi
Trends Beachfront Guesthouse -- another nice newish place just to your right as you get off the boat, in front of The Sand Box.
Iguana Reef Inn: The "Ritz Carlton" of Caye Caulker -- very nice suites for US$70 to $90.
I agree that The Sandbox is the best restaurant on the island -- big portions and moderate prices. Other good places: Cindy's and Glenda's for breakfast, Syd's, Wish Willy's, YooHoo Deli for a quick sandwich.
The smell of the water (and it's not unique to Tropical Star) comes from sulphur in the water from shallow wells. Quite a few places have water that smells, but some are worse than others. In general you should drink bottled water on Caulker.
Belize First Magazine http://www.turq.com/belizefirst/