I practiced cutting a few more bottles yesterday. Trying to refine my technique.
I decided that rather than risk thermal shock to the entire bottle I would pre-warm the bottle by filling it with hot water. I filled it up and let it sit for a few minutes to come to temperature.
I then scored it and ran the score line under the cool water from the faucet. I then watched as the score line became thicker as it fractured through the bottle. I did need to reheat the score line with more hot water, and repeat by running under the cool water.
I was able to get a fairly clean cut of the bottle. I ended up with only one irregularity in the cut.
To smooth this out I used emery cloth to reduce the size of the jag, and to dull the edge of the bottle.
Once the rim of the bottle was relatively flat, I moved on to the grinding phase. I used silicon carbide powder on a piece of glass. I dropped some on the glass and used a water bottle to wet it.
Then I ground the glass for about 15 minutes, re-wetting and adding additional silicon carbide as necessary.
I verified lightly with the emery cloth that the edges were not sharp, and then carefully checked on bare skin.