My basic setup was vim/emacs (only as text editors) + grep/sed. It is totally doable and it is possible to be really productive but makes no sense when you have actually tried out adding some integrated tooling to your editor.
Depends on the company (and country). At least here in Stockholm (SE) take home coding assignments and/or whiteboard problem solving seems pretty common for both junior and senior positions.
Sign with the other company then resign from the first. This is not uncommon. How ever do it now before starting and taking any relocation assistance from the first company.