votalite is more suitable in this context. The word has the meaning of 2-face.

volatile
virtuous
facetious

The clue for first two blanks is rude and explosive character. What kind of person will be of this character, that person will be volatile and ignoble. And how the career of such a person will be? It will be dangerous. So, precarious will be the third blank’s perfect fit.

The lawyer’s explosive, rude remarks convinced many that he was (i) _______ and of (ii) _______ character, suddenly making his future as a politician seem (iii) _______ .

Since the character is decided by his explosive and rude remarks, volatile - which means quickly evaporating and hence almost explosive is the word for the first blank. For the second blank, we can eliminate courageous and virtuous, both of which are positive qualities, and hence ignoble is the correct choice.

Since a politician has to be careful with his words and be measured in discourse, in order to survive the vicissitudes of political life, somebody bursting out rudely is going to be doing poorly as a politician. He will be endangering his political career with that sort of attitude. Hence precarious is the word for the third blank.

precarious - not securely held or in position; dangerously likely to fall or collapse.

Volatile, Ignoble , Precarious

Volatile is suitable since "explosive" is an adjective used to describe the lawyer.

Such a shitty lawyer! I feel bad for you
I have an exciting story that happened with my lawyer friend and me. My lawyer friend is 12 years older than me. We are excellent friends since childhood. When I graduated from high school https://www.vocationaltraininghq.com/how-to-become/paralegal/, he took me as his assistant. When we were approached by a family who wanted a divorce, we had a case, but they could not divide the acquired property. As a result, after the trial, the judge decided that they should divide everything in half. The guy who turned to us took a chainsaw and cut all the property in half, ruining absolutely everything. His ex-wife was shocked when she saw all the things cut in half and decided to sue him, but the court rejected her request to pay her compensation because her husband had good lawyers.

1. If he made rude remarks out of the blue, he would be considered volatile. Hence, C.
2. Although it could be considered courageous to make rude marks outspokenly, it's probably more likely to be ignoble -- not honorable. Typically, you don't want to make outspokenly rude remarks in court; thus, it would be considered dishonorable; hence, F.
3. If he's rude and ignoble, it doesn't make a whole lot of sense for his future to be guaranteed. Facetious is beside the point. Precarious -- reflecting uncertainty -- fits best. Hence, H.

C, F, H.