because you would have to say "good to see you too!" but you can just say the last 2 words "...you too". to make it shorter :)

there are many peculiarities (odd things) with respect to english phrases. Example : does he drive a sports car? doesn't he drive a sports car? Answering 'Yes' to both questions means the same thing, even though the second question is technically different. But that is English today. I think Russain speakers may find this hard to understand.

me too would be good to see myself too

Well forget about it, people always screw that up, there's not much logic here. It looks like "you too" is more popular in daily speaking, while you'll find the formal reply should be "me too" according to formal references.