Is failing a verb

Yes. Failing is the present participle of fail.

It is used with a be verb to make continuous tenses eg:

past continuous -- He was failing to to do his job properly.

present continuous -- They are failing to keep up a good standard.

Failing can also be an adjective: -- She received failinggrades.

Failing can also be a noun: -- They were quick to point out the plans failings.