1. The word each is refering to "languages", meaning that the languages they studied took a few months for every language.
2. With his father means that they both held responsibility, but the wording is vague. The amount of responsibility depends on context.
3. egg is an adjective in this case, and is describing what the cup is meant to hold. The noun is cup.
4. Someone left this note to you is not grammatically correct. Something closer to "someone left this note 'addressed' to you" which would just give a distinction of a name written on the note. Both would be practically the same.
5. Correct, those sentences have different structure, but the same meaning.