In addition to Jason's comments, when $g\geq 2$, $\overline M_{g,0}(X,0)=\overline M_{g}\times X$. The obstruction bundle is given by $\mathbb E^\vee \boxtimes TX$ where $\mathbb E$ is the hodge bundle. Then I believe a splitting principle computation plus a little intersection theory on $\overline M_g$ should give you the result. When $g=1$ you need one marking, and the computation follows from the similar idea.

Not sure whether you get notifications from comments in other answers or not. To be sure I just repeat some messages here. I really appreciate all your answers here. Ultimately we want to simplify a bigger expression that includes the RHS of my question 2 as a special case. It's pretty long and currently still over my head. I don't know if you have the time and the interest to take a look. But in case you do, you are welcomed to send me an email (in my profile).

Would you mind telling me your real name by email? Because if we decide to post anything about this work in the future, we will acknowledge you (also Fedor and other people). I just temporarily added my email in my MO profile. We are still working on an ultimate combinatorial expression that includes all my questions as special cases. It's too long to post in a comment, but if you are interested, I will be glad to send it to you by email as well. Of course interests from other people will also be welcomed, just let me know in the comment or send me an email.

Nice! I really learned a lot from all the answers and appreciate everybody's effort. In particular, this generating function method seems to have some other applications in our project. I got another quick question. Let $H_n=\sum\limits_{k=1}^n 1/k$. Do you know whether we can similarly describe the formal series $\sum_{n\geq 1} \dfrac{H_nn^n}{n!}z^n$ using functional equations just like your $T(z)$ and others?

Cool! I actually updated my question to include a more general identity that I want to prove. I kept thinking there is some generating function lurking behind, and I'm just looking at the coefficients. Any thoughts will be greatly appreciated.

Sorry about my ignorance and thank you for pointing out the right words! I am thinking about asking around, and your comment definitely helps.

Thank you! This is a really cool argument, especially for me who doesn't have a lot of experiences in graph theory. I will mark this as the correct answer, but of course any further ideas and comments about question 2 are welcomed as well.