Would this be a reasonable idea for math classes?

As long as you are proficient at algebra, trigonometry should not be that challenging. The basic notions are simple and if you can memorize a few identities and how to derive other identities from these, you should be OK.

Trig is usually a prerequisite for Calc 1 and most students don't do well because of weak Algebra or Trig skills, so if you need Trig for Calc 1, why take Trig after Calc 1? It doesn't seem logical to me. Unless of course the university you are going into doesn't have Trig as a prerequisite and you can take Trig anytime you want. Either way, in my opinion, if you are going to learn some Trig in your free time before Calc 1, you might as well take the whole Trig course to be more well-rounded.

So, I would take Trig in the summer from your community college to get a more solid foundation or fall at the university instead of Calc 1. Your cousin can still give you a head-start on Trig to make Trig easier for you. Calc 1 during the summer is usually not easy or any generally hard course for that matter, especially if you say you are not gifted at math. At my university, there is a fairly high percentage of engineering students failing Calc I, Calc 2, and Calc-based Physic courses in the summer. Try to take these kind of courses during the fall or spring. Use RateMyProfessor.com to avoid professors that may be too challenging for you. When it comes to where to take Calc 1 and Calc 2, I recommend asking a university adviser about that because some degree programs require you to take a certain number of credits/hours (whatever system they use) for prerequisites at the university. I would say most universities are open-minded to letting students take electives like humanities at community colleges. Probably not so much if the courses are important for your degree program. Check the university's website for adviser contact information or go visit the academic advising office in-person.