You should absolutely apply for unemployment, no matter what. There is nothing to lose. You fill out the info and then your previous employer has to dispute the claim, in writing. If they don't, it's smooth sailing. If they do, then you'll be asked to fill out additional paperwork where you can state your case.
One thing many people don't realize is that while it is true that employers can fire you for pretty much anything (if you live in a state where the law is "at will,") it doesn't mean your termination is a good reason to deny you unemployment benefits, though it can.
A big key is going to be how much proof your employer can provide that you were previously warned about these actions. The Department of Labor really likes to see that the employer gave you a fair chance to correct what you were doing, so long as it's not a major offense (theft for example), which it sounds like you're clear of. From what you have said, the verbal warnings seem pretty meaningless, especially if it didn't come from a figure of authority. They are also harder to prove, but they may have kept a record of it. The written warning is sort of damning, but the fact that it didn't come from your boss is odd. And if that guy previously asked you to help with other employees, that's really strange. The problem is that will be hard to prove and it's sort of their word against yours.
Another important component will be...the company policy (or lack there of) about what you did. Hopefully you still have an employee handbook to refer to.
Even if you lose round 1 and the DOL rules in the company's favor, you can appeal it and force the company to participate in a hearing and someone else will then hear your case, which could be good. Your company find it isn't worth their time to continue. If you lose that, you can appeal it once more and then it goes in front of a panel. I'd advise trying to fight for it to see if you can get it. There is no penalty for being rejected unemployment.
A lot of companies will try to make up reasons to jettison you, but if it's a good company, and they know you didn't do anything wrong, they should just let the unemployment claim go. A lot companies do.