Learn Turkish: Kim yorgun değil?

The question word kim (who) acts like a "he/she/it" (third-person singular) subject. Because the third-person singular doesn't take a personal suffix, değil stays in its plain, base form.

Could I use a negative suffix like -me or -ma instead of değil to say "not tired"?

No. The suffixes -me and -ma are only used to negate action verbs (like "don't go" or "not running"). Because yorgun (tired) is an adjective, you must use the separate word değil to make it negative.

Where is the word for "is" in this sentence?

Turkish doesn't have a direct translation for the word "is" used in this way. The "is" meaning is built into the grammar of the sentence, and değil essentially handles the entire "is not" meaning here.

Why is değil at the very end of the sentence?

In Turkish, the main action, state of being, or its negative marker almost always goes at the end of a sentence. So instead of "Who is not tired?", the Turkish structure literally reads as "Who tired not?" (Kim yorgun değil?).