How do I hang a large wall to wall mirror?
I'm sure there are several ways to secure your mirror, However the two basic methods are 1) mechanical side clips and 2) Mirror adhesive. 

The safe and easiest way is to use mirror clips that screw into the wall around the parameter of the mirror. Small drywall anchors are normally used if you're not lucky enough to have a stud along the horizontal sides. Try to find a stud along the top or again, use anchors.  Be sure to read the ratings on the clips to determine how many you need for the weight of your mirror.  It is really your call if you want a gap under your mirror. personally, I would leave and inch or two for cleaning purposes. A mirror resting on the back splash may collect water and dirt.

The second method is to use a special mirror adhesive.  This makes for a very clean install, but can be very tricky. Read the directions on the adhesive completely. Be sure you have a firm finish on your wall to apply the glue to. Some side or top supports may be needed to hold mirror in place until the adhesive has cured.  Once you use this method, it is semi permanent as it will normally destroy the sheetrock when removed. CAUTION!!! Only use a mirror adhesive if the mirror manufacturer recommends that method as an option. Some mirrors will not glue up and the reflective silver layer will pull off and allow the mirror to come crashing down!