I believe "Would you like a put of tae?" would be:

Ar mhaith leat cupan tae?

First of all, am I correct so far? Secondly, to say "If you would like a cup of tea, please come inside" would be:

Dá ba mhaith leat cupan tae, tar isteach

How close am I, or any alternate suggestions.

cupán (síneadh fada)

dá ba - in Standard Irish: dá mba.

But ... it isn't really an irreal condition, so I'd think: Má bá mhaith leat ...

I Ros na Rún deirtear "ag iarraidh tae?"

Yes, in Conamara we use "ag iarraidh" for "wanting".

An bhfuil tú ag iarraidh tae? Do you want tea?

And we don't usually include "cupán", as you're more likely to get a mug anyhow

A Bhríd, would you say in general Ros na Rún is a good example of colloquial Connemara Irish?

If the character is a native speaker like Thadhg, then yes. And when Séamus and Peadar were there.
But I've lost touch with it for the last few years so I don't really know who the other characters are now so I can't say.