Shanon, If you purchase laminate material, it's usually in sheets, so you've already got a good edge, to butt-up to a surface. If you need to cut the material before gluing, a router is the way to go. You'll probably get some chip-out, unless you support the material well, or move slowly - have never had great success with tape, always gummed-up my router bit. Since you'll need extra material around the edges, chip-out isn't a big worry, as the chips are usually very small (<1/16"). The final pass, with a laminate-trimming router bit, will clean-up the edge very nicely (the bit has a bearing, just make sure it rotates easily).
As for your other question, about the thickness of the countertop, that's a preference issue. I've seen 1/4" pressed wood used (it was well supported), all the way-up to 1". I guess I would go with at least a 1/2". It comes-down to how well supported, how easy to manage (size/shape), what type of application (the sides of a cabinet don't need to be as sturdy as a bench-top), and how much you're willing to spend (it's not a big difference between 1/2 and 3/4 material, but on a big job, it adds up)?
Henry posted a good response to a similar question on July 28th, at: http://www.renovateyourworld.com/wwwboard/messages/9380.html
Here's another link that explains the entire process: http://www.housenet.com/iq.asp?categoryid=102&sub_categoryid=1&detail_categoryid=115&articleid=1915