Plumbing – Do narrow utility sinks exist

That isn't how gravity works.

Drains like sinks are typically gravity drains where the water is drawn down into the drainpipe... by gravity.

This is in contrast to things like washing-machines where the wastewater is pumped out into standpipes.

In order to drain your sink via a standpipe above the fixture you'll need a drain pump or drain into a sump and pump it out from there.

There is no specific limit on the number of sequential studs, as the 7/8" limit is intended to reduce the potential stresses, however, given the choice, I would rather drill than notch:

2308.9.11 Bored holes. A hole not greater in diameter than 40 percent of the stud width is permitted to be bored in any wood stud. [...]

and

In no case shall the edge of the bored hole be nearer than 5/8 inch (15.9 mm) to the edge of the stud.

So 40% of 3.5 in gives you ~1 3/8" to use, often much more than you need for supply lines.

My concern though, is that you're thinking of notching for a waste line, and notching 7/8" to add a 1 1/2 or 2" drain line is... I'll just say that I'm with @JohnSmith in that I wouldn't bother, but then I don't know your specific situation.