To your latter point, any time we do technical exercises, we _always_ let the interviewee use their own IDE. Nowadays it's easy because most people have laptops, though we do offer to set one up if they do not have a laptop to bring.

Their choice of toolsets and setup can also be a really interesting discussion topic. More than once during an interview, someone has shown me something really cool that I adopted (new key binding, a new tool, a particular way someone had their terminals split-paned, etc).