“Jane’s going to the movies” is correct. The apostrophe indicates a missing letter or letters, and goes where the missing letter goes. So “Jane is” becomes “Jane’s”, just like “do not” becomes “don’t” and “cannot” becomes “can’t”.
You’re right in the first place. “Jane is going to the movies” would be contracted to “Jane’s going to the movies.” Furthermore, “Jane has gone to the movies” would be contracted to “Jane’s gone to the movies.” So “Jane’s” can be either the possessive of “Jane” or a contraction of “Jane is” or a contraction of “Jane has.” The reason that this doesn’t look right to you is that you don’t often see noun-verb contractions in print, while you do see pronoun-verb contractions fairly often in print. Even though in speech people often make noun-verb contractions, they tend not to write them even when trying to reproduce their speech.
Can’t we quibble a little? Surely we can find something. After all, it’s the Fourth of July and all. (I changed that from “Independence Day” so that I wouldn’t be exclusive.)
I know that it is grammatical to use a possessive pronoun before a gerund as in:
I admire your walking to work every day. (“Your” instead of “you”)
What happens when a proper name is substituted for "your’?
I admire (Caroline, Caroline’s) walking to work every day
Both of those’re correct. But they have different meanings. “I admire Caroline walking to work every day” means you’re admiring the person. Maybe she’s good-looking. “I admire Caroline’s walking to work every day” means you’re admiring the act. Maybe she’s determined.