How can I run a cable past a horizontal block between studs in my wall?
You need a long, flexible auger bit. I have this model and found it invaluable when I need to add a new cable inside the wall. The kit has a 90 degree handle that fits on the shaft and lets you bend it to fit through the wall opening to reach the bottom or top of the stud bay and drill through. This forum page gives a nice rundown and pictures of the bit in use.

There's a eyelet in the bit where you can fasten a cable to pull back through the hole you drilled, too.

The mid-bay 2x4 is a brace. Some walls have them, some don't. I don't know whether that's a builder's decision or building code. None of my walls have them.