What does 「うちえあそびにきませんか」mean?
My notes say that it means "won't you come to my house to relax / to have a chat?" but its literal meaning is "won't you come to my house to play?" and that sounds really dirty. I don't want to unknowingly make inappropriate requests in Japanese.

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あそぶ means to have a good time, to spend time with someone. 
あそびにきて is a set phrase asking someone to come to my house, to visit my place (to chat). It doesn't mention what to do.

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In addition to what hana-san said, "うちえ" is actually a mistake. It should be written "うちへ". The pronunciation is the same but you write it differently when it is the particle "e" (to, at, etc).