A three year old has vomited three times within 10 hoursafter the first dental cleaning and fluoride treatment. Could the dental appointment have caused it?

Highly unlikely. It is highly unlikely the vomiting was related to the dental appointment. Any vomiting 10 hours later would most probably not be related. Fluoride can make some kids vomit if the taste disagrees with them or if they swallow a sufficient amount - but that happens immediately. The illness was most probably coincidental.
Doubtful. I see above that the child in question is 3 years old and on wellbutrin (bupropion)...Is that correct? I would really like to know why, if that is true. That medicine should not be used now. The dental cleaning should not have caused the issue, unless the child accidentally swallowed the prophy paste, and/or the Fluoride varnish/gel. Even if that did happen, i still doubt that is what caused the vomitus.
Probably not. It's possible for children to swallow too much toothpaste during the cleaning and then patients will have an upset stomach and possible vomitting. Given the amount of vomitting that occurred, it would seem unlikely for it to be from the dental cleaning.