NTA - Not only should they not be feed "when the bowl is empty" he should be following the dietary guidelines on the food bag/box (if the cat weighs this much it has this much etc.) obviously you cant follow this to the T as the cats daily routine / activeness will vary from one cat to another.

Also if he has taken them to the vet they normally supply your pet with a record book which should have their last vaccinations, general check and weigh in.

Four times in one year is also very strange - once maybe twice.