1. Wrong: While I dont know who's car's she had last week or earlier, I do know shes borrowing Haydens this week.
Better: While I don't know whose cars she had last week or earlier, I do know she's borrowing Hayden's this week.
Explanation: Contractions (don't, she's) need apostrophes. Possessives (Hayden's) also need apostrophes. Possessive pronouns (whose) don't. Non-possessive plurals (car's) don't use apostrophes.
2. Wrong: At Riley's and Jessie's party, people talked about what state's they're from.
Better: At Riley and Jessie's party, people talked about what states they're from.
Explanation: Riley and Jessie seem to be having one co-hosted party, so the first name doesn't need the possessive. States, which is a non-possessive plural, needs no apostrophe.
3. Wrong: Who do you think cobbled Jesuses sandal's?
Better: Who do you think cobbled Jesus's sandals?
Explanation: Possessives need apostrophes while plurals don't.
4. Wrong: You can find the womens' room slightly closer to the pool than the mens'.
Better: You can find the women's room slightly closer to the pool than the men's.
Explanation: In possessive plurals, the apostrophe goes after the original, plural word, which does not necessarily mean that it goes after the s.