is "me" before sinto necessary? "Às vezes eu sinto triste" without "me" sounds ok to me.
Sentir-se is a reflexive verb. In formal Portuguese it's used (with the reflexive pronoun "me") to say "I feel sad." In colloquial BrP, it is likely to be omitted.
I would say it's necessary. At least that's how it is in Spanish. It's just the way it is.
Many languages have this, "I me feel," "I anger myself," etc. To a native English speaker, this can often sound very weird, and occasionally pervy (I feel myself... excuse me?!).
This is a contraction. In Portuguese they call it "crase".
a (preposition) + a (article feminine singular) = à
a (preposition) + as (article feminine plural) = às
yes, gotta have it or you answer (like I did) "the times I feel sad" as if it was just "as vezes" rather than "at the times"