Does rim width affect what tire size I can use

The answer is "yes". The tire manufacturer will have a recommended minimum rim width for each tire size. These are shown in the specs on the Tire Rack website, for example.

From experience, 185 is fine on the 4.5" rim... probably 205 as well. 215 is OK on the 5.5" rim — I have 215/70-14 on my rally wheels.