For a 1-2 month stay there are a few options,
Hidden Garden Cabins have quite a good monthly rate of $300
Seagrape Plantation has budget rooms with shared kitchen and bathroom for $250
or you can try to find an apartment there are a few around.
As for dive equipment - Coconut Tree Divers has a good stock of Oceanic and DiveRite equipment, there is also Sueno Del Mar and TGI divers stocking Mares, Scubabpro and other makes. Coconut Tree's prices are comparable with US prices, but can in no way compete with the likes of leisure pro.
As for shore diving - most shops will give you a fill for $5, I do not know anyone who will rent you a tank - so bring your own! Most of the diving is very close to the shore but entry and exit can be difficult since there is a lot of iron shore or just limited access to the shore. Basically very few people shore dive anywhere other than at the AIs.
When you are around that long I would recommend getting very friendly with a dive shop at you may find they let you dive for virtually nothing. A good trick may be to try and do your AI with Marc at Coconut Tree - it will cost $200 and then they are likely to occasionally use you as a dive guide, DSD confined water skills etc. on very busy days in return for $5/dives.