Is it normal for your toenail to fall off after haveing ingrown toenail surgery?

It can. Any time you stress out a toenail with surgery or even a small in-office procedure, there's a small risk of the nail falling off completely. Nails don't tend to take a joke very well, and if they're angry enough after a nail procedure, they may just give up and start over. (this is a very rare occurrence, though.).