The answer to both of your questions is yes. This is because, one, people can sue for pretty much any reason at all (in your first example, it would likely be a privacy/publicity rights suit) and, two, publishing someone's copyrighted material without permission from the copyright holder is considered copyright infringement.
Whether anyone could win an invasion of privacy/publicity rights suit or a copyright infringement suit based on what you have presented here is a question only a court can answer, and the court would base their answer on the particular photos showcased on the particular websites, and on all of the facts presented surrounding the taking and publishing of these photos. An attorney who can review the photos and websites in question can give a better idea of what the odds are of any lawsuit being successful.
A defense for a photographer in any invasion of privacy action, where no written consent forms were signed or releases obtained, is that the taking of the photos was done with the "implied consent" of those featured in the photos. Implied consent can be argued if the subjects of the photos were aware that photos were being shot and did not at the time express any objection to being photographed. This also holds true for any photos taken of the instructor, who apparently was aware of the photo-taking and allowed photographs to be taken in his studio. Another argument for the photographer that could be a potential defense to any invasion of privacy action is that the subjects pictured were only "incidental" to the photographs as a whole and not singled out as the focus of any one photo.
An argument against implied consent is that minors cannot legally grant consent, implied or otherwise. Permission from a parent or guardian is necessary. It is never wise for a photographer to take photos of and/or publish photos of other people's children without the consent of the parents or guardians of these children. There are risks. And, if the instructor or a student other than the photographer's son is featured prominently in the photo(s), the instructor or student would not be incidental to the shot but the focus of the shot. To use such a photo for advertising purposes (ie. to advertise a new photography business) would be in violation of publicity rights laws.
A defense for the martial arts instructor in any copyright infringement action brought against him by the photographer, for posting the copyrighted photos taken of him at his studio, would also be that there was implied consent to use the photos. The photographer sent him the photos with no conditions outlined as to their use. Another potential defense would be under the "de minimis use" doctrine, where the use of the copyrighted material is considered trivial and inconsequential. Whether a use is de minimis or not, however, is for a court to determine.
An argument against these potential defenses for the martial arts instructor is that copyrighted material was used without the express written consent of the copyright holder. If the photos can be seen as an advertisement for the martial arts studio, then the photographer may be entitled to compensation for the use of his photos.
Implied consent, and any oral consent, is often hard to prove and can make for a difficult defense, which is why a wise photographer will have releases signed prior to taking photographs of identifiable people, and especially prior to the publication of photos with identifiable people as the subject. Signed consent forms become necessary if identifiable people in any photo are to be used in any form of advertising.
Although an invasion of privacy/publicity rights action, or a copyright infringement action, would be unlikely due to the costs involved in pursuing either, the photographer may wish to pull his photos of the instructor and the students until he is able to get releases from those pictured (especially if they are singled out in a photo and not merely incidental to the shots), and he may wish to grant a limited license to the instructor to use the photo(s) of the instructor on the martial arts website with the condition that the photographs of the martial arts instructor that are displayed are credited to him. And the martial arts instructor, likewise, should pull the copyrighted photos from his website until such a license to use the photos is granted.
