Learn Italian: Signore, attento, sta per cadere! - Sir, careful, you are about to fall!

The structure stare per followed by an infinitive verb (like cadere) is used to describe an action that is going to happen immediately. It translates directly to "to be about to" do something.

Why is the verb sta used instead of stai if the speaker is talking directly to the man?

Because the speaker addresses the man as Signore (Sir), they are using the formal "you" (Lei). The formal "you" uses the third-person singular verb form, which is sta. If they were warning a friend, it would be stai per cadere.

Could I say attenzione instead of attento here?

Yes! Both work well as a warning. Attento is an adjective meaning "careful", so its ending changes depending on who you are warning (it would be attenta to a woman). Attenzione is a noun meaning "attention" and never changes its ending.

Why is there no article before Signore, like Il signore?

When you are talking directly to someone to get their attention, you drop the article. You only use il signore when you are talking about him to someone else (for example, Il signore sta per cadere means "The gentleman is about to fall").