Simple Share Screen Question

Hi

My Mom just got a new mac mini and we are trying to get share audio and share screen to work.

1. If I take my mac mini to her house, and plug it into her modem/router (it's a modem with four ethernet ports), is this "the same" as if we were communicating via internet miles away? If not, is it still useful for getting the share audio/screen to work?

2. We used to share audio/screen via ichat, now it's via imessage, right?

THANKS!