Learn Italian: Pur costando molto, il viale sta venendo ampliato. - Even though it costs a lot, the avenue is being expanded.

This is the progressive passive. It combines a conjugated form of form of stare (to show ongoing action) with venendo (the gerund of venire, used here as the passive auxiliary) and the past participle ampliato. Italian uses venire in these progressive structures to emphasize that a dynamic action is happening to the subject right now.

Could we say "sta essendo ampliato" instead?

Technically yes, but native speakers almost always avoid sta essendo. Combining stare with the gerund of essere sounds very clunky in Italian. Using venendo is the standard, natural-sounding way to express that something "is being" done.

What exactly does "pur costando" mean, and could I just use "anche se"?

Pur followed by a gerund like costando is a great way to say "even though it costs" or "despite costing". You absolutely could say anche se costa molto, but pur costando is slightly more elegant and very common. Keep in mind that you can only use this pur + gerund structure when both halves of the sentence share the exact same subject (in this case, il viale).

Why does ampliato end in -o here?