Do a layout before you start.  Arrange your layout so you don't have sliver pieces and you should be fine.  I would snap a few lines to keep your first row on layout.

Also take a look at the tile to see if it's a patterned tile or random.  If random, then you may want to checker board the tile.  

Follow the layout suggestions above.

Clean is the key word here. Excess wax, soaps or grime will diminish the quality of your efforts, but yes, you can go over the the top of the existing flooring. This is assuming that the suface is smooth. Repair any holes or other concave imperfections with a good filler, I use Bondo, as it is tougher that ALL of the products made for this purpose, yet a little bit more exspensive.

If there is doubt about the cleanliness of the surface, use a strong soap and maybe some type of abrasive, or even TSP, which will actually etch the surface a little, and help your results.

OdT