Can a butcher block countertop be rounded with a router?
Any common butcher block countertop can be cut and rounded to fit.  You'll have to refinish the sections you've cut, but there are no other special considerations.  Cheap tools are often OK for one time jobs, there are also tool libraries where you can rent or borrow better tools.  Some home stores offer rentals.

For finishing butcher blocks, the traditional materials are some form of food safe oil.  For a desk you have the option of sanding and using any sort of furniture finish from a wax to a varnish to polyurethane.  It's best if you buy the butcher block unfinished, so you don't have to match finish.  An excellent discussion (focused on food uses) is at http://www.thewoodwhisperer.com/articles/cutting-board-finish/