Run plumbing in the wall. there is some framing in the way. Can I cut it out

Removing plaster can make tremendous amounts of dust, isolate the area with plastic to cut down on the spread, and get yourself a dust mask and a helmet is a good idea. Cover any cold air returns and heating vents. Once you're prepared, taking plaster walls down is pretty easy, whack it with a hammer to bust a hole, then you can usually get a shovel or pitchfork behind the plaster and it will pull away from the lathe easily as you break the keyway that holds it in. Clean up all the plaster and dust, then go after the lathe with a crowbar and/or hammers; then start cutting the vertical studs with a sawzall.

Double, triple, quadruple check that it's not load-bearing, if the blade is binding when cutting through a stud, there's downward pressure, and even if it's not structurally load bearing, things could've shifted over time above, and taking it out might make things drop or sag.

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.