I'm not a native italian speaker but I am a gardener. The answer is yes. A major division is between fagiolo rampicante (pole beans) and fagiolo nano (bush beans). All beans are legumes. Many further divisions are possible along the lines you suggest, e.g., fagiolo nero, fagiolo bianco, and specific varieties such as fagioli cannelini or geographic denominators such as fagioli di Sarconi.
G as in mangi.
As a general rule, "gi" and "ge" are ALWAYS pronounced as "G", as in "Gina", respectively "George", no matter where you find them in a word (beginning, middle or end).
Same goes for "ci" and "ce" which always have the "tch" sound, as in "Czech" or "Tchaikovsky".
I am not aware of any exceptions whatsoever. If any, they would be only on words of foreign origin (most likely just names and places).
If you want to have the "g" as in "gut" or "c" as in "cut" sounds, you need to add an "h": ghe, ghi, che or chi. Which by the way is the only function the "h" has in Italian when following a G or a C.
Side note: if you ever fancy learning Romanian too, the above pronunciation rules are identical and specific to these two language only (some similar rules in other Romance languages, but not really the same). :)