Are there any definite places to stay for someone who is under 18 in New York City? I cannot go with anyone over 18 soon enough, and want to investigate the college campuses (NYU and Columbia)

Are there any actual places to stay? Hostels? Hotels?
And hopefully you can provide actual places, not just "probably"s.

Anything is appreciated, though.

As for hotels, call a few and ask about policy. Perhaps if your parent were to make the reservation and secure it with their credit card it wouldn't be a problem. As for hostels, I found this list:

The Seafarers and International House
123 East 15th Street (between Irving Place and 3rd Avenue)
1-212-677-4800
[www.sihnyc.org]...
Room: $$

Hosteling Int'l NY
891 Amsterdam Av at West 103rd Street
1-212-932-2300
[www.hinewyork.org]
around $35 a night and up

Jazz On the Park
36 West 106th Street (between Manhattan Avenue and Central Park West)
1-212-932-1600
[www.jazzonthepark.com]
starts around $30 a night

Central Park Hostel
19 West 103rd Street (corner of Central Park West and West 103rd Street)
1-212-678-0491
[www.centralparkhostel.com]
$30 at night and up

The Big Apple Hostel
119 West 45th Street (between 6th and 7th Avenues)
1-212-302-2603
[www.bigapplehostel.com]
$35 at night and up

I have no personal knowledge about any of the hostels.

You might also contact NYU and Columbia about accommodations. Sometimes schools will let visitors stay in a dorm and shadow a student. They also may be a resource for other accommodations as well.

Good luck!