Learn Turkish: Dün gece forumda saçmalayıp duruyor muydu? - Was he continuously talking nonsense on the forum last night?

This is the core of the -(y)Ip durmak structure you are learning. While durmak usually means 'to stop' or 'to stand,' when you combine it with another verb ending in -(y)Ip, it creates the meaning of doing that action repeatedly or continuously. You can think of it as 'standing firm' in doing an action.

What is the root verb in saçmalayıp, and how is the suffix attached?

The root verb is saçmalamak (to talk nonsense), which comes from the adjective saçma (absurd/nonsense). Since the stem saçmala- ends in a vowel, we need the buffer letter -y- before adding the 4-way harmony -ıp suffix, giving us saçmalayıp.

How is the 'was he...?' question formed here with muydu?

To ask a past continuous question ('Was he doing...?'), you use the present continuous ending -ıyor on the main verb (duruyor), and then attach the past tense to the separate question particle. mu (question) + y (buffer) + du (past tense 'he was') gives us muydu.

Could I just say 'Dün gece forumda saçmalıyor muydu' instead?

Yes, but it changes the nuance. While saçmalıyor muydu simply means 'Was he talking nonsense?', adding the -(y)Ip durmak structure emphasizes that he kept on doing it relentlessly. It often adds a feeling of annoyance, which fits perfectly when talking about someone spamming or ranting on a forum!