I do this. I think this applies only to adverbs modifying verbs. Adverbs modifying adjectives or participles stay put.

"She runs quick," is a thing I'd say.

"***The topic was hot debated," would be ungrammatical to my ears.

Not sure how widespread it is. I think it just falls out of a natural tendency to elide utterances which don't alter the meaning of a sentence. In many positions it's obvious that an adjective is meant to modify the verb rather than a noun.

It's not a hard and fast rule. In formal writing I'd use adjectives per standard grammar.

Maybe also related to the (standard) use of adjectives as describing the state into which something is transformed by a verb. In "I painted the wall red," "red" is properly an adjective and modifies the transformative act, not the object. I suspect this construction has been unconsciously widened to apply to nontransformative verbs also.

Notably "***she quick runs" sounds highly ungrammatical to my ears.

Flashbacks to my driver's ed teacher in highschool who would say adverbs correctly then correct himself by saying it again without the -ly. This drove me insane. And no I haven't noticed this generally in US english speakers. I would assume some negative things about people who drop the -ly from adverbs.

Can you give an example? I've never noticed that (except for certain specific dialects and slang) but I may be blind to it.

It's correct if enough people do it and do it consistently.