Faucet is secured in a strange way. How can I remove it?
This looks like a faucet that is meant to be installed from the top without having to get under it to tighten nuts.  The pieces underneath that are holding in the holes look like they rotate like a toggle bolt.  Remove the handles and all the trim from the top and then look for a nut or screw that will turn to loosen/lower the locking pieces underneath.  Once it is loosened enough, the bottom pieces should rotate to be in line with the valve bodies and should pull out of the hole easily.

I've never seen a faucet like this in person, so this is an educated guess, but I'm pretty confident.