Possibly the houses foundations have shifted and now the bottom of the sink is sloping the wrong way.
The fix for this would be releveling the house, the cabinetry, the benchtop, or the sink.
There are coating substances used for relining porcelain-on-steel bathtubs, I don't know if they are durable enough for a sink, or how well they can fill low spots.
Sure, you could fill it with an epoxy, or you could tile it.
Note: Only since you don't care about aesthetics, since I suspect either will look awful in short order, if not right out of the gate. My mind's eye makes a beautiful tile sink bottom (like a shower floor) but the real world experience part of my mind doubts that the detailing at the edges could ...
A guideline is at most 4 inches from the lip of the counter to the front of the sink. This has to do with sink access for little people.
The back does not have to equal the front, but you need to keep space for your faucets of course. The issue is mostly the clearance required by the faucet, underneath. Usually that's another 3 inches or more, but read the ...
Just above the upper threads is a black cap. Insert a flat screwdriver slightly between the upper treads and the cap and pry it up. Then use a coin to unscrew the diverter. Replacement part number is A66G391, which comes with a removal/installation tool.