Could be a part of a dialogue:
-- Скільки вам/тобі років? - 35. - Мені теж! (How old are you? - 35. - Me too!)
Or part of a problem:
Мені це не подобається. Мені теж (I do not like it. --- ???)
I grew up using the word також and this is the first i've seen теж. Are both common and acceptable forms?
"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)
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."