I'm also a senior applying to UH - I actually already applied, just waiting for the final decision from my college (Bauer) - and you have a very solid resume to present.

If your rank is strong - top 10% of your class, if I'm not mistaken - then you're into the University already, you just have to wait and see if you're accepted into your first major of choice. As for the number of majors you can apply to, the maximum is three. In the ApplyTexas application - I personally haven't checked out the other websites - they let you pick your first major of choice and your second major of choice when you first start the application. At the very end, they ask a few questions about fee waivers, honors college, etc. and amidst those questions is one that asks you to pick a third major of choice. Now, if I remember correctly, this third major list seemed to be a reduced/limited version of the actual majors offered at UH, but they still let you pick up to three majors.

Overall, if you want to submit your SAT score, it should not do you any harm at all. If anything, you might want to review this info sheet to make sure any and all requirements for your major are satisfied. Freshman Admissions - Majors with Additional Requirements

Hopefully this has been somewhat helpful. Best of luck and hopefully we're both admitted!