1. 'Me too' is always correct. It's an all-purpose response that you can use following any positive comment.
'I'm bored' .....'Me too'
'I like this place' ....... 'Me too'
'I passed my exam' ..... 'Me too.'
'I might go home now' .... 'Me too'.
NB Note that you can't respond to a grammatically negative comment with 'Me too'. If the comment is negative, such as 'I don't like this coffee', the response is 'Me neither'.
2. You would not say 'I'm too' using the contracted form. You can say 'I am, too', using the full form of the verb, but ONLY if the comment you are responding to contained the present tense of the verb 'to be'. For example:
'I'm bored' ......'I am, too'.
'I'm from Paris'..... I am, too.'
'My friends are going to the beach' .... I am, too'
For all other verbs, you need to repeat the auxiliary e.g 'I can, too', 'I was, too', 'I do, too' and so on. Native speakers do this instinctively in conversation, like a game of ping-pong, but non-natives may find it tricky. So, if in doubt, it's easier to stick with either 'Me too' or 'Me neither'.