"Get on (well, badly, etc)" is perfectly fine. The thing is, certain verbs sit comfortably with a plural subject. If the subject is single, then you add the other person using "with". eg. They always argue over politics. She always argues with her father over politics. Did they break up? Did he break up with her? The same goes for "get on" (and "get along"): Jack got on well with his uncle Fester. Jack and (his uncle) Fester got on well. They got on well. I'd actually say that, when the subject is plural, the verb is intransitive (reflexive). Adding "with..." just fills in the missing information.

I am going to assume you meant " Jack got along well with his Uncle Fester." I'll tell you that since "with" is the preposition it connects both what precedes and what comes after. To answer your question with forms a unit with the noun phrase " his Uncle Fester." This is my answer because you are describing who Jack gets along with. To answer your second question the verb "get along" or "get on" as you put it is actually a phrasal verb and a transitive verb.

I think "get on with" is British English, so I'm not certain how they use it. However, according to Oxford's EFL dictionary, it's also okay to say "Uncle Fester and I got on well." In that case, I think "with" belongs to the noun, and not the phrasal (multi-word) verb. http://www.oxfordlearnersdictionaries.com/definition/english/get#get__789