Learn Russian: Кто ни кричит из зала, я буду выступать. - Whoever shouts from the hall, I will perform.

When you combine a question word like кто (who) with the particle ни, it creates a concessive meaning equivalent to '-ever' in English or 'no matter who'. Кто ни кричит means 'whoever shouts' or 'no matter who shouts.' If you used не, it would simply negate the verb, so кто не кричит would mean 'who doesn't shout'.

Since this lesson focuses on несмотря на, how would I express this same idea using that preposition?

Because несмотря на (despite) must be followed by a noun or pronoun in the accusative case, you would need to slightly rephrase the sentence. You could use a noun, like Несмотря на крики из зала (Despite shouts from the hall), or use a linking phrase: Несмотря на то, кто кричит из зала (Despite who shouts from the hall).

Why is из зала used for 'from the hall' instead of от зала or с зала?

Prepositions of location work in set pairs. To say you are 'in the hall,' you use в зале. The standard opposite of в (in/into) for movement out of an enclosed space is из (out of/from). От means moving away from the outside of something, and с is the opposite of на (on/off).

Why is the verb выступать in the imperfective aspect here?