I am a fluent French speaker, and when we speak of age, we use a verb for "to have."

In our case, it's avoir. For the 1st person Je, it would be conjugated as ai, so to say how old I am, I'd say I have X years: J'ai X ans.

I wonder if you use iru, to have, in the same way?

No, we don't do things. Rather, one normally says

何歳ですか　(Nansai desu ka?)

You can also say,
年はいくつですか　（Toshi wa ikutsu desu ka?)

These sentences are in polite speech.

Anyways, in both sentences there was a counter phrase.

The verb in these sentences would be the copula.

Is the sai counter a verb ilk?

Counter phrases can be considered nouns or adverbs depending on the context. As for age, it is going to be nominal in that context. Counters simply count the number of something. There is a section in Lesson 7 about this.