Would it be correct to say: "He should have gone past that place, turned right, and gone to the pub." (when explaining that someone went the wrong way)? The verbs are homogeneous and thus I believe supposed to appear in the same form, but for some reason I have a feeling that it should be "...and went to the pub". I'm struggling to find the rule or at least examples of similar sentences.
This is an example of a parallel structure, where repeated parts of successive clauses are omitted. Here's what it looks like fully expanded:
He should have gone past that place,
[He should have] turned right, and
[He should have] gone to the pub.
It is now clear that the final clause cannot be
[He should have] went to the pub
“Went” is the past and is also used as a past participle, but less often than “gone”. This means that “went” in your example would be grammatical but it does not mean that it would be optimal.
Your example has the structure “He should have A, B, and C. A, B and C are best as identical constructs, each using the past participle. “gone” is conventionally and more often used than “went”.
Consider “ He should have gone past that place, turned right, gone to the pub, and drunk a beer". This reads better than the slightly confusing mixed past and participle sentence “He should have gone past that place, turned right, went to the pub, and drank a beer”