Plumbing – Is it a problem that the toilet flange is 3/8″ below the surrounding tile

The toilet flange needs to be on top of the finished floor. Meaning the bottom edge of the flange needs to be on the same plane as the toilet. So if your toilet sits on the tile, the flange needs to be on top of the tile too. The spacing of the toilet exit "horn" and sealing surface is designed for this height.

almost all plumbers agree, here are some sources:

http://www.terrylove.com/forums/showthread.php?51976-Toilet-flange-height

if you have it lower than the finished floor you risk leaking toilet water (the water is the best part) into the floor or worse, in-between the tile, sub-floor, backer board. This is because the wax ring was designed only to take up the air gap, not to provide a "tunnel" or extension of the pipe/flange/toilet horn.

A diamond hole saw will do the job, use plenty of water and light pressure.

You will do better with a single hole, AFA breakage goes, compared to making 2 edge semi circles.

Update

If we are talking about a toilet flange, there are 2 methods, each with variations, depending on your equipment.

1. With a tile wet saw, raise and lower the blade such that you cut diameter lines edge to edge. Rotate the tile 10 degrees and repeat. Flip the tile over and repeat from below. Remove teeth with an angle grinder fitted with a diamond blade. If you are connected to a GFCI outlet, a helper can slowly spritz the cutting with a water bottle. It will keep the dust to a minimum. I would recommend a continuous edge (not segmented or stepped) diamond blade for this. The cheap table type tile wet saws actually excel for this cut, allowing fast and accurate cuts, all from below.
2. The whole cut can be made with an angle grinder setup described above, you just have to tip the blade at 45 to the surface of the tile and sneak up on the line from the inside of the desired cut out. With marble you have to go slow and minimize pressure. This method also works nicely for electrical boxes.