What are the IRC requirments for bracing/blocking in a load-bearing 2×6 stud wall that is 12′ high

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.

You can install wood blocking between metal studs, but you'll have to open up more of the wall to attach them (you obviously can't toe nail them). Notch the block so it fits into the channels in the studs, then just attach it with screws going through the studs.

I wouldn't hang a sink on tile without some sort of additional support. Even if the tile were strong enough to hold it, it would provide a point of leverage against the wall that could introduce flexing.