If you have extra glass that didn't fracture off cleanly, you can usually break it off without a problem. If your crack extends down into the cup, you'll either have to discard it or re-score the bottle lower for a shorter cup... or at that point, a mini-bowl. Maybe a dish for soy sauce when you have sushi...
As long as you polish out most of the cracks and chips (stress points), your cup should be relatively safe in the dishwasher. Go with the top rack though.
The bottle I used was for Woodchuck Hard Cider, scored and cut 4 inches from the base. It comfortable holds 1 cup of beverage, with about half an inch of clearance from the top.
If you didn't see my original bottle-cutting video, click the link in the first line. If you haven't seen the details about my bottle-cutting jig, click here.
Options for future variations:
1. Use a thicker bottle. Even with a rounded lip, it feels weird to drink from because we are all used to drinking from mugs with substantially thicker walls. If you frequently drink from shot glasses or plastic cups however, you'll be right at home.
2. Etch the bottle/cup for a unique, classier finish and look.