Learn Russian: Осторожно, пол очень скользкий, смотри, не упади! - Careful, the floor is very slippery, watch out, don't fall!

You are right that negative commands normally require the imperfective aspect, which would be не падай. However, when warning someone about a sudden, accidental action they should avoid, Russian uses the perfective aspect. You are warning them to prevent an accident, not telling them to stop a deliberate action.

How exactly does смотри work in this sentence?

Literally, it means 'look', but here it is used as a warning phrase like 'watch out' or 'make sure'. Combined with не and a perfective verb, the set pattern смотри, не... translates to 'make sure you do not...' or 'watch out so you do not...'.

The current lesson mentions using the Perfective Future for warnings, but isn't упади an imperative (command) form?

Yes, good catch! Упади is the perfective imperative. When making warnings with смотри, не, Russian speakers interchangeably use either the perfective imperative (смотри, не упади) or the perfective future (смотри, не упадёшь). Both mean exactly the same thing in this context.

Why does скользкий have an adjective ending if осторожно ends like an adverb?