It's likely that piping will have to be adjusted, especially if you add a garbage disposal. Besides, all this is done in PVC piping these days

You're stuck because the horizontal pipe can't be lowered below it's current position (where it attaches to the p-trap). If you have an undermount sink, you could replace it with a drop-in sink that could be a bit deeper because it would its edge would be above the countertop instead of below it.

Is that a double bowl?

Yes, it is a double bowl, drop in sink.

How thick is your current countertop