After the surgical removal of an avm from the brain, can it come back? If so, how likely is it?

Unlikely. If the avm was completely removed ie. The nidus removed and all feeding vessels that are not feeding other essential ares of the brain are clipped, the likelihood of the avm returning is slim. If the entire avm could not be removed due to its location, there is a chance it would again recruit additional blood supply and grow. There is therapy to address this, talk to your neurosurgeon.
Unlikely if removed. Surgical removal of the avm leads to obliteration of the nidus. If there is angiographic demonstration that this is gone, then it is not likely that it will reconstitute and form new vessels. Bear in mind, that avms are congenital lesion. Thus, it is something you are born with and they will not grow de novo.
Not likely. Avm's are congenital lesions that, when resected, should not recur.
Low. Most surgeons will perform a blood vessel study called a catheter angiogram to confirm complete avm removal after surgery. As long as this is negative for residual, the chance of recurrence is extremely low.