You are legally correct about the Tea party not being a formal party but it would be more about running under a label as opposed to a party. After all isn’t the GOP nothing more than a mere label anymore? It’s become a fools party. What does the old fool Mitch stand for? And then there’s the young fool Ryan. GOP leadership is so stagnant that the people are starving for label that means something, even I it’s single issue such as “build a wall”.
No. It is actually a distinct legal entity that gets access to ballots for it's nominees, etc.