A vector has no fixed size — that is the point of it. So pick the width you actually need and it is drawn crisply at that size, rather than at whatever default your browser happens to assume.
This page loads no libraries at all — the work is done by the browser itself. Your file is read on your own machine and never sent anywhere — open your browser's network tab and watch while you use it.
Whatever width you set, with the height following the drawing’s proportions. Leave it at 0 to use the size written in the file.
Yes — PNG keeps it. Convert to JPG instead and transparent areas become white.
A screenshot is stuck at screen resolution. This redraws the vector at your chosen size, so it stays sharp however large you go.
Everything here is free and works the same way — in your browser.