Is it customary to build "built-ins" over carpet?
You should cut away carpet.  The cabinets will last longer than the carpet and will just be a mess when it's time to install new carpet.  I'd want the cabinets secured directly to the floor, not through carpet/pad.  

When you pull up the carpet and baseboard, you'll see the tack strip.  If you are careful, you can re-use it, but it is cheap to replace. You'll probably want a tool called a knee kicker carpet stretcher to properly re-attach the carpet to the tack strip.  You can rent them or cobble one together yourself. 

Edit:
It looks like you are working on a slab.  In which case you'll need special tack strip with concrete nails and removing the old tack strip will be more of a chore.

In a pinch, or where the carpet is hidden from view, I have used large headed button-head screws.