“Is it possible to lose your ability to talk after neurosurgery?”

Yes, it can happen. Generally, the speech areas are avoided in surgery, but sometimes they cannot be, for example, if the person has had a bleed in that area or a tumor growing in that area. It can also happen if the speech areas are retracted to get to a deeper lesion, but the effects on speech are usually transient in such cases. Thankfully, it doesn't happen often. When patients undergo surgery in this area, they are usually warned of the risks.

Yes

There are several important areas of the brain related to speech. For most people, these area on on the left side. Some left-handed individuals have speech centers on the right side. When a procedure occurs near these areas, great care is generally taken to prevent speech problems through pre-surgical planning as well as during surgery. Sometimes this even requires doing the surgery with the patient awake. If there is physical damage to one of these areas, or a stroke that happens, speech could be affected. This does not happen often, but the risk is very variable depending on the skill and experience of the surgeon.

After brain surgery, there is certainly a risk of neurologic deficits occurring. While they not be common, they certainly happen. It depends on where in the brain the surgery was being done, the type of surgery, and in some cases complications (for example bleeding) that can occur hours after surgery. Speech impairment is one of the problems that can happen

"Not talking" requires some further explanation. There are speech areas in the brain (usually on the left side in right handed people) which are speech related. Interference with these can lead to a variety of temporary or permanent speech problems ranging from problems naming things, to difficulty with fluency to totally garbled speech. There are other areas of the brain that can lead to mutism (no speech at all, notably in an area called the supplementary motor area or in the cerebellum). Fortunately mutism from surgery in these areas is almost always reversible although this may take weeks to months. In primary speech areas we can do awake surgery and stimulate the brain to see where speech is to avoid trouble. This is necessary because no two people are alike with regard to where speech is although usually in the same general area.

Yes and it depends on what part of the brain was worked on. There is a speech center normally on the left side in the temporal lobe region. There can be damage done that can affect the ability to talk and or understand what someone else is saying.

Yes, it can happen, depending upon where the surgery is occurring. It does not happen often.