Can I install an under-mount sink in an existing Formica counter?
If the substrate under the formica (which is a thin veneer) is particle board, you must do something about the exposed edge of particleboard around the hole that they originally cut to put the sink into.  Not only is it ugly, but it will suck up water and expand and disintegrate.

There are kits to glue a curbing over this edge that purport to waterproof it. I am VERY leery that such a thing can work over the long term.  Water just gets in - that's its job.

Your fixtures must be mounted through an apron of the sink that extends to the back?  They would remain so, and just extend up from below counter level (there is probably no countertop underneath them now.  If you look underneath, you will probably see that the hole for the sink goes back further than the fixtures).  This will create a cleaning nightmare :(

The hole will probably need to be neatened up (it's probably not a nice even shape) and it is probably a bit bigger than it needs to be.  The sink will be held to the underside of the counter by clips that screw into the countertop from underneath, and you will use silicone on the rim of the sink before lifting it into place to provide a seal (and probably run a fillet around the joint after it's in place, wiping the excess away with your finger).

It sounds like a lot of work, and a lot of things that can go wrong.  I'd suggest just keeping it overmount.