I am interested in applying for a K-1 visa for my fiancee in Japan (I'm living in NY). Because I am a full-time masters student, I make enough money to support two people but I do not have enough assets to show for it (i.e. large adjusted gross income numbers). Therefore, I am interested in having a co-sponsor. My cousin in California has agreed to do this, being that he makes enough for his own family plus the amount for two people at 125% of poverty level (this is the required amount I have to prove that I possess). What are all the responsibilities of a co-sponsor? Also, is it okay that my co-sponsor lives in another state? Lastly, I may start my PhD soon and my top choice is a school in California. If I move to California for school, am I allowed to even apply for the K-1 visa being that I will not have lived there for long? Must I stay in NY in order to apply for this particular visa? I appreciate any advice on this, because government-sponsored websites do not give much information.
YOU are not applying for a visa, your fiancee is.
You may petition for her, which gives her 'permission' to apply for a visa. There is no requirement about how long you need to live in a district to qualify to petition.
There is NO financial requirement for the petition.
At the visa application/interview, your fiancee will need to show that she will not become a Public Charge. Normally this is done by the US fiancee completing the Affidavit of Support I-134.
If your income is less than $17500 GROSS/year, the additional question becomes: How are the two of you going to support yourselves? (I'm not looking for the answer from you; it's a rhetorical Q).
If you have a co-sponsor, like your cousin, his income must be enough for his family/household + your fiancee. This may be more than what you've calculated.
You should also look ahead to the part --after-- the visa.. Adjustment of Status, which will cost an additional $1500 or so and requires a more stringent Affidavit of Support, the I-864.
This is a good site for explaining things in regular speak:
http://www.familybasedimmigration.com/forum/index.php