How to know if I deserved to be promoted?
I suggest that this question may not be the right question to be asking. What one "deserves" depends on the point of view: your manager's, your peers', yours, etc. (For instance, my mom has always thought I deserve to be President...so what? Maybe I should have sold more real estate...but I digress). What really matters is not what you deserve but what you actually end up getting. We can all probably think of managers who in our eyes definitely did not deserve to have any authority over other human beings. But someone else thought differently, and here they are. Bottom line: if you are promoted, that means you 'deserved' it in some shape or form, from the perspective of those in authority to promote you. That's really all that matters and there is no point mulling this over. You are now a lead. Great. It happened. Now just do your best being one, and don't lose sleep over whether you deserved it in the first place.
maybe they promoted me to team lead because my manager didn't have another option and couldn't wait for one?
Maybe. Or maybe not. Or maybe both. Or neither. Again, thinking about this won't help you, your peers, or your manager. This is a classic type of issue that is really, really worth doing your best to just let go trying to rationalize, and instead to focus on doing the work, each day at a time. If it helps, keep in mind that change always ignites speculation, initially. But just when you think it all through and feel like you got everything organized in your head, you discover that the world has moved on and that other people have other problems to occupy themselves with, and it's water under the bridge. Was it worth all the agonizing? In hindsight, probably not.
I know I have it in me. But who doesn't?
I can easily think of some people who don't, and you probably can, too. I am sensing deeper issues of self-esteem, and need for approval and validation, which may be driving the constant tendency to compare oneself with others. Listen, there is always going to be someone who deserves to be in your shoes more than you (or so you, or they, or their mothers, think). But guess what, the same is the case about them. It works both ways. So, it really doesn't matter. I would recommend to work on your self-appraisal and to try to move away from constant comparison, and toward finding that inner 'center' which allows you to find happiness and joy in each day regardless of what others think (or may think!) of you.
Did you threaten them with another job offer and they promoted you?
People are people, they say odd stuff all the time. In this case the obvious implication is, "did you only get promoted because you threatened with leaving, rather than business need, your supervisory ability, or manager's good judgment?" Such thoughts cross just about everyone's mind. But some people do a much better job keeping such stuff inside, and quietly resolve these inner doubts. Others let it all out, and leave you to deal with their projected insecurities. The point is, their insecurity should not become your problem. Work on building up your internal psychological walls to prevent their issues from spilling over and consuming your mental and emotional energy. Learn to distance yourself from such questions and thoughts. Humor is a great tool for this. So is calling them on it: "Yeah, you know, I fugured the ONLY way I would ever get promoted is to threaten them, rather than try to earn it. Obviously that worked! You should try it."
But what about the others who think I wasn't deserving of the promotion? How do I deal with their comments and questions?
See above. Hopefully some of these ideas and strategies will help you deal with these questions and comments. Good luck!