If I take my license exam in California, but my residence is in New York, would my call sign still reflect the New York location?
When you fill out the paperwork at the time of taking the test you specify your permanent home address for your license. Thus, if you take the test in California but use a NY address then you will be assigned a call sign appropriate for your license level for New York which uses the 2 zone.
Licenses like for a Technician license are issued usually as a 2x3 call sign and the suffix usually is assigned in alphabetic order. So, for example, you could be issued something like a KG2XYZ call sign. I am not sure about the 'G' part of the prefix or where the suffix falls but there are web sites where you can find out what that is because they list the most recently issued licenses for each zone number.
If you want though, after licensed you can request a Vanity call sign and it can be any available and legal US call sign you choose. A friend I know here in the state of Washington requested a Vanity callsign in the 0 zone because the suffix he wanted was his three kids first name initials. So he is a zero-callsign in Washington state (zone 7).
If you apply for a more advanced license, you can possibly get a 1x3 call sign or maybe even a 2x2 call sign.
In case you are unaware, the different call signs are known by how many letters are before the zone number and how many after. So, a 1x3 call sign is like mine, K7PEH. A friend of mine has a 2x2, or KE7FB. And, another friend of mine is a 1x2, W7IZ. A 2x3 is something like KD7ZVB or whatever.