I have two questions:

First, can student on a student VISA get federal student loans?

Second, I am calculating out my expenses and assuming i can get federal student loan, I will need to max out the yearly limits for the Stafford and PLUS loans. So, I will also need to take out private loans. Should I apply to many different financial institutions as well to increase the likelihood that my loan will be approved? Or will applying to many financial institutions actually lower my credit score?

Thanks in advance.

Yes, you can get federal loans on a visa. Next, you should talk to the financial aide department people at your school, they will tell you who will approve you and guide you through it. Most of these people are real good at getting money. You have to remember, the more money they get you, the more you will spend at the school, so they have an incentive to help.

My cousins that are studying here from Spain applyied for and received financial aid for international students. They did not qualify for the typical financial aid.

There is very little financial aid for foreign nationals to study in the US, with the possible exception of citizens of Canada and Mexico. Most grants, scholarships, and loans from public and private sources are restricted to US citizens.

However, check out this link to a site that lists the sources of loans for foreign nationals.
http://www.edupass.org/finaid/loans.phtml

you could consider private loans. many of those are willing to lend to people abroad, but they do come with a higher rate of interest. but the euro is strong right now anyway.