I have never seen a sink with the sort of problem(s) you describe. It sounds like it has a thin or cheap material as a surface....maybe both. A good sink will have a classic porcelain finish and should last longer than YOU.

The splashing problem 'might' be the design of the spout itself. There are a wide variety of faucets...high, low, angled...etc.

I would ask friends or neighbors if THEY have any such problems and if possible, look at any brand names. If possible, take pics to compare when you go shopping. Home Depot will not only have a variety, but they often have experienced former plumbers on staff who might shed more light on your problem.

There 'are' ways to coat old tubs and sinks with a hard overcoat, but that is usually done just when one doesn't want to rip out a tub.