You could rephrase the English as, "How does he feel about himself?". The "si" is the reflexive bit that makes this question different from "How does he feel/hear something else". I'm not a native speaker, but this is how I understand it:
Il veterinario sente un tumore sul gatto. Come sente? Sente con i mani.
Il veterinario lavora sodo. Come si sente? Stanco!
The first one presents 2 problems:
1 - There is no subject, and "sentirsi" is already reflexive. So you have to make impersonal a verb that already contains the particle "si". You have to use an additional "ci" for this purpose. Then, it becomes "Come CI si sente?"
2 - When verbs are used as complements and follow the main verb, you have to leave the "to" in the infinitive. Remember you say "how does it feel to put...?", in english, not "how does it feel put...?". Italian "to" is "a".
Putting all together, your first sentence becomes "Come CI si sente A mettere la mano nel fuoco?".
'come si sente uno mettendo la mano nel fuoco'? is ok, you could say "come si sente uno a mettere la mano nel fuoco?", alternatively.