In the key of A, for example, I have the A chord, and the Bm and Cm chord.. but the Cm chord is C, D#, G i believe, and the actual note 'C' is not in the key of A, so how can you play the chord if it has a note outside of the key?

The key of A has C#m, not Cm (which is C Eb G). You can & should experiment with playing chords that have notes outside of the key though, if it sounds good thats all that really matters
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Stash Jam is totally right. It is technically a C#m, but there are too many songs to even name that play a III chord as a major, probalby the most common one being E - G - A. A lot of blues progressions play those kinds of chords in sequence, all major. There is a G# in an E major chord but no G, but if you play this, it still sounds good.
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