'Me too' would usually be the end of a statement unless you follow it up with a conjunction, i.e. "yeah, me too, but....", but if you are speaking for a group of people it would be more common to hear "We ______ as well", in a sentence like "I play football", "We do as well". In regards to which is grammatically correct, 'we too' is your best bet but would sound a bit old fashioned in conversation. And for third person singular and plural, I would use 'He/She/They also', i.e. "He/She/They also play/s football", but my previous comment about 'we too' applies here as well.

It depends on whether the word is acting as a subject or an object. He would like some cake. "We, too, would like some cake." (Or, "he, too, would like some cake.") We or He is the subject. I gave him some cake. "Give some to us, too." (Or "give some to him, too.") Us or Him is the object.

We might say "me too", "us too", "him too", or "you too". We normally use objective case for this expression. For positives, we say "me too". For negatives, we say "me either" or "me neither". I like it! Me too! I don't like it! Me either!