If a school asks for a classic/pre-1965 song as well as a contemporary song, would it be bad for the pre-1960 song to be belted or sung in chest voice? (If the contemporary was also belted). For example–and these are not my songs but just random–if someone were to go in with If I Were A Bell from Guys and Dolls and sing that with a chestier sound and for their contemporary they sang Astonishing from Little Women. Would it be better to sing something like (again, not my auditon song, just an example) Mister Snow from Carousel instead of If I Were A Bell, to show head voice as well as belting? Please share your opinions on this, thank you 
I think showing off a range of things you can currently do well vocally (as well as an actor) is preferable. If you do not yet have a solid head voice, or are not yet a belter, I would not sing songs where that is the expectation. However, if you can find songs that show off a range of what you can do as a singer and actor that is a win win.
Agree with Kat that if you can show a range of what you are capable of singing, that is the most ideal.
Anecdotally…when my D auditioned for BFA in MT programs, she didn’t feel strong enough at the higher range of the legit MT voice. She was still working on developing it and not fully confident with that range. She auditioned at age 16. Her strength was (still is) the belt voice. The way it turned out, I recall two BFA programs vocalized her to get a clearer picture of her range. Of these two, one accepted her and one Priority Waitlisted her. At another school, U of Michigan, which did not vocalize her at the audition, she was able to get feedback after her denial. She was told that she did not show enough of the legit soprano voice. In any case, she had a successful audition season, getting into a number of well regarded programs and landed at her first choice. Freshman year of college, the entire year in vocal performance classes was spent on the legit voice and she surely developed a higher range that year.