The first phrase is correct. It is the most precise and elegant way to say what you mean in this case. Some people might use the second phrase but to me it's not correct English.

Yes, we use both phrases, and I'd understand them to mean the same. The second is more unusual, but it certainly makes sense in the context. In fact, I don't think that a native speaker would be likely to say 'He photographs well' to mean that he is good at taking photos. A foreign learner might use this phrase to mean that, but it might sound unnatural coming from a native speaker. Another natural-sounding phrase (which IS ambiguous) is 'He takes a good photograph'. If it's clear from the context, this can also mean that he looks good in photographs. This phrase and 'He photographs well' are both nice examples of ergative verbs, if anyone is interested.

1 is fine :) 2 is also fine I suppose, or you could say he or she is good at photography.

I'm a U.S. native speaker. I WOULD say "he photographs well." And, to mean, the meaning is unambiguous. If I mean "he is a good photographer" that's what I'd say. I would only use "photogenic" in a joking way. To me "photogenic," used seriously, belongs to the world of professional models and professional photographers. In ordinary life it's out of place. To say "my friend Susan is photogenic" would be like saying "Coca-Cola has a wonderful bouquet with fruity notes of citrus," or "Taylor Swift has an exquisite portamento." If a baby were in the middle of spitting up I might say "He's not very photogenic right now."

I use both but with a subtle difference (personal taste, not official): 1. "He/She is photogenic" implies that the person is good looking both in person and in photos. 2. "He/She photographs well" implies that the person looks better in photos than in person.