Learn Hardcore Dutch: De bus is weg.

Weg here means gone/away: it states that the bus is no longer present. It’s not the noun de weg (the road). You can tell because there is no article before weg, and it follows is. The noun meaning needs an article: de weg.

Examples:

De bus is weg. = The bus is gone.
De weg is smal. = The road is narrow.

What part of speech is weg in this sentence?

It functions as a predicative complement (often treated as an adjective/adverb). It does not inflect:

De bussen zijn weg. (not wege)

Similar words used this way: kwijt (lost), klaar (ready), stuk (broken).

Why is it is and not gaat or heeft?

Because it describes a state. Dutch uses zijn (is) + a predicative to express states.

De bus is weg. = The bus is gone (state/result).
De bus gaat weg. = The bus is going away (action in progress or about to happen).
De bus is vertrokken. = The bus has departed (focus on the departure event). Using heeft here is ungrammatical.

How do I say it more explicitly as “The bus has left/departed”?