"I hate Jill singing those songs." = "I hate Jill when she is singing those songs."?
Solution 1:
I would interpret them differently.
"I hate Jill singing those songs" implies that you hate her actions (singing) when she sings those songs.
Whereas "I hate Jill when she is singing those songs" emphasises that you hate her (Jill) when she sings rather than hating her singing.
Solution 2:
"I hate Jill singing those songs" could only be interpreted as meaning something like "I hate it when Jill sings those songs", "I hate that Jill sings those songs", or ""I hate Jill's singing those songs". I agree with Chris H's answer that "I hate Jill singing those songs" could not practically be interpreted as meaning "I hate Jill when she is singing those songs."
I'm not sure yet of the exact explanation for why this interpretation is ruled out. It does seem to be possible to interpret the similarly constructed sentence "I hate the girl singing those songs" as meaning "I hate the girl who is singing those songs".
I think it may be related to the phenomenon called "whiz-deletion". In some cases, postnominal participles are analyzed as having some kind of relationship to wh-relative clauses using a form of the auxiliary be.
When we look at similar sentences with explicit integrated relative clauses of this form, "I hate the girl who is singing those songs" is fine, but "I hate Jill who is singing those songs" doesn't seem right.
Something seems to be going on with information structure here: in a sentence like "I hate the girl who is singing those songs", the clause "who is singing these songs" is what would traditionally be called a "restrictive" relative clause, in that it communicates some information about "which girl" the speaker is talking about, whereas in a sentence with a proper noun not preceded by a definite article like "I hate Jill...", the proper noun by itself completely specifies which person we are talking about, which seems to preclude the use of an integrated relative clause or of related postpositive constructions that can be analyzed as being derived from relative clauses by "whiz-deletion".
For example, I have a similar feeling about the sentence "I hate the girl in the green hat" (it seems OK, and means "I hate the girl who is in the green hat") vs. "I hate Jill in the green hat" (it sounds odd, and feels like it might mean something like "I hate it when Jill is in the green hat" or "I hate having Jill in the green hat").
Solution 3:
In typical use, you can rule out this interpretation. Reading Jill singing those songs as a noun phrase is much more natural than reading it as Jill (when she is) singing those songs.
On reading or hearing that sentence your audience would understand your hatred to be of the action not the individual. If you wanted to give the other meaning you would have to make explicit the when she sings those songs (or ...is singing...)