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Мене is accusative (as someone else mentioned.) Accusative is used for the direct object of a verb and movement to places after prepositions. Direct object is the thing or person being acted upon. For example "ти любиш мене." "You love me." I am being loved, so I get the accusative. Dative is used for indirect objects and certain set phrases like this one. Indirect objects are often preceded by "to" in English, like "He gave a book TO ME."
"Теж" can always be substituted with "також", but not vice versa. (Я теж хочу їсти = Я також хочу їсти) Також is also used in a conjunction "а також" (and also): Я люблю їсти яблука і груші, а також їздити на велосипеді (I like to eat apples and pears, and also (I like to) ride a bike)