Can a DVI monitor connection offer a higher resolution than my current VGA connection?

The monitor is what determines the maximum resolution, not the method in which it is connected. If the maximum hardware specification of your monitor is 1400×1050 then it isn’t going to change with the use of a different cable. DVI can provide a better picture then VGA, if you are interested in why you can check this link.

Almost certainly. You’ll want to get a high quality DVI cable, but unless you have UXGA or the like, most DVI cables will be better. It’s important to get the right kind, however. Check here for the lowdown on DVI Cables.