How close to the shul would a Rabbi generally require congregants or a prospective convert to move (in kilometers please!)?
Well, what's reasonably walkable? Probably about 2 kilometers or so.
Another significant factor -- if the synagogue's neighborhood has an eruv, it's reasonable for a rabbi to expect people to move within the eruv -- it will be far easier to observe shabbos if you can carry in the neighborhood (especially if most of the locals are used to doing so). Take Pittsburgh's eruv, for instance. You could find points within a 2-km radius of the synagogue that are outside of the eruv.
Most eruv maps are up online these days, so search around a bit (or ask) and you should be able to figure it out.