The case of compound sentences, [independent clause 1] + and / or / ... + [independent clause 2], has already been well covered on ELU. See ect's balanced answer at comma before 'and' in compound sentences – necessary or not?. The comma is optional, more appropriate in some examples than in others.
Here, compound predicates (examples [2] and [4] above) will be looked at. Deletion of repeated subject leaving compound predicates is well known.
This is probably going to end up classed as a recommendation rather than a rule.
(1) If the parts of the compound predicate involved are semantically closely connected (eg causally, or especially those corresponding to phase structures), and especially if they are short, a comma is better omitted after say 'and':
She tripped and hurt her knee.
He sat and watched.
A comma is not forbidden here, but would signal an unlikely pause in reading. With the second example above, the phase structure interpretation [he sat watching] would be discouraged: 'sit' could well now be punctive 'sit down'.
(2) With less closely connected parts of the compound predicate, especially when weighty, a comma is useful in parsing:
- She went to the most disreputable club in the most disreputable part of town, and began handing out leaflets inviting people to the upcoming church fete.
This is especially true with more complicated sentences:
- He took his time selecting exactly which shirt to wear to the party, taking great care, and chose the one his girlfriend had bought him the previous Christmas, the one with the baroque design.