You can say "φαίνεσαι καλός" without μου and "μου φαίνεσαι καλός". It's actually the same because when someone seems fine, he seems fine to you actually, and it's your personal view. But both are used in Greek so I believe "to me" was put in the answer so that you can remember this way of saying that, too. Μου may add some kind of "moderation" (sry if this word is wrong in this context) to what you say because you emphasize that this is your opinion.
It refers to health, but it is an adverb. For example: "The elephant is well" (well applies to is, shows the state that it is in) whereas "the elephant is good" (good applies to elephant, not the state that it is in, but a characteristic of it).
EDIT: I just noticed that "well" is actually an adjective in English when refering to health. Consider the explanation above as what is true in Greek and how well=καλά is used. It seems that you learn something everyday! Thanks for pointing that out @JamesTWils! Have a lingot from me! :) (It still seems weird to me how "well" can be an adjective in "I don't feel very well", though :P)
We don't tend to use adverbs to modify the verb "to be" in English, though an adverb could be used to modify an adjective following "to be," e.g. I feel terrible or I feel terribly ill, but not I feel terribly. In most other Germanic languages, an adjective can essentially be used adjectivally, so there's really no difference. That used to be the case in English as well, I believe.
No, because the verb is φαίνομαι, φαίνεσαι, φαίνεται, φαινόμαστε, φαίνεστε, φαίνονται. Here, the (implied) subject is 'you' (sing.).
Check this discussion's comments for a detailed explanation about weak-strong pronoun forms and then think of the above sentence as "Εσύ φαίνεσαι καλός σε εμένα". Does the structure make more sense now?
The difference is that φαίνεσαι καλός and φαίνεσαι καλά have different meanings in Greek. Φαίνεσαι καλός can only mean "you look good" and not "you look well". That's because καλός is an adjective and "well" is an adverb, as you said. The Greek sentence means that you seem to be good, not that you are well.
On a string of losses here! My third try, this time with "You appear to me to be good," also wasn't accepted. The answer given, "You appear good to me," is ambiguous in English (in a way I assume the Greek sentence isn't) between "You appear (to all the world) to be good to me, but not to others" and "You appear to me to be good (in general)." I guess I'm in need of some hand-holding here.