So, me too! --- no need to put "So" here

Here, the word "So" means "thusly" or "in this manner". When you respond "So do I" you are saying "I must also act in this manner." The word "do" is actually a verb saying that your are performing the same action as the person you are responding to. "So me too". There is no verb showing action.

The first one is correct, but the second one isn't. You can say, "So, me too." You just say, "Me too" OR "Yes, me too".

I would say the second set, "So me too!" is an error in diction. The first phrase, "So do I!" is correct. Also, remove the comma.