I'm going to start by apologizing to Andrew. I think I kind of derailed his answer. But I'm going to attempt to explain it. I do hope you read the other article about actual idiomatic speech that Ronald Sole linked, because since your question seems based in very formal prescriptive grammar, I'm going to ignore more idiomatic grammar, and base my answer on purely prescriptive grammar.
So, I believe the short rule here is when an -ing verb is a participle or a gerund, it becomes the new linguistic head of the phrase. That means everything after it has to conform to its grammatical and syntactic rules (and nothing before it really matters). In comparisons, participles and gerunds have this grammatical form:
-ing participle or gerund
+ (more or less)
+ adjective or adverb
+ auxiliary verb
playing games more than I do
talking less than he does
I think it's also possibly important to discuss the difference between the following:
I like going to the store more than he does.
I like my going to the store more than his going to the store.
The first one is the simple rule I told you about how comparison usually works after -ing participles or gerunds. The second example has two parallel gerund clauses, but the meaning is very different. The two different meanings, respectively, are these:
I like to go to the store more than he likes to go to the store.
I like it when I go to the store more than I like it when he goes to the store.
I don't know if that previous example helps at all, but I think these different meanings might help explain why comparisons work a bit differently when an -ing participle or gerund is the head of the phrase.
Note that infinitives also use this structure: I would like him to read more than she does. Idiomatically, many speakers might say I would like him to read more than her. However, formally speaking, the infinitive is the new head of the phrase and should take the nominative. There's also a risk of ambiguity. You can't tell for sure which of the following things "I would like him to read more than her" means (formally speaking, it means the second one, but most speakers would mean the first one):
I would like him to read more than she reads.
I would like him to read more than I would like her to read.
On a final note, you said the following didn't sound right to you:
He resents your being more popular than he.
It sounds pretty weird to most native English speakers, as well, because nobody says it, anymore, but formally, it is completely proper. We are at kind of a weird place in English. There are still some people who are uncomfortable with saying something like this (and at least as many more who have no problem with it):
She is taller than him.
However, the people who are uncomfortable with that, generally speaking, no longer ellide the word "is" and would say:
She is taller than he is.
Finally, while saying the following is formally correct, it is very rare:
She is taller than he.