Learn Turkish: Dün o asabi adam durmaksızın yakınıyor muydu? - Was that irritable man complaining continuously yesterday?

It comes from the verb durmak (to stop). The suffix -maksızın or -meksizin is an adverbial suffix meaning 'without doing something'. So durmaksızın means 'without stopping' or 'continuously'. It is a slightly more advanced and emphatic equivalent to the -madan suffix you already know, such as in durmadan.

Why is the question particle mu placed in the middle of the verb in yakınıyor muydu?

In Turkish compound tenses like the past continuous (was doing), the question particle always comes after the main tense suffix (-ıyor) but before the past tense copula (-du). Breaking it down, you get yakınıyor (is complaining) + mu (question particle) + y (buffer consonant) + du (past marker).

I learned the word sinirli for angry or nervous. How is asabi different here?

While sinirli can mean someone is currently angry or just has an angry personality, asabi (an Arabic loanword) specifically describes a permanent character trait. An asabi adam is a short-tempered, irritable, or high-strung man by nature.

I usually use şikayet etmek for 'to complain'. Can I use it here instead of yakınmak?