From a syntactic point of view, I would not call this dropping a pronoun—I would call it conjoining verb phrases.
The word "and" can join various different types of units:
I fed the dogs and the cats.
I fed the dogs and pet the cats.
I fed the dogs and Helen pet the cats.
In the first case it's joining noun phrases; in the last case it's joining entire clauses.
So what's it joining in the second case? I would say the verb phrase: a unit made up of the verb and its object(s), but not its subject.
Why analyze it this way? Well, look at what happens when it's an object rather than a subject that's repeated.
I fed, and Helen pet, the cats.
The commas there indicate a noticeable pause in speech, indicating the place where the repeated object has been elided. We don't see this pause when it's the subject that's repeated, indicating that something different is going on in the subject case.