Can you play different chords at the same time?
In music and music theory, a polychord consists of two or more chords, one on top of the other. In shorthand they are written with the top chord above a line and the bottom chord below, for example F upon C: FC. The use of polychords may suggest bitonality or polytonality. Harmonic parallelism may suggest bichords.
Do chords change in different keys?
When you play in a different key, C major's role changes. If you're playing in the key of C major, then C major is the root chord - the I. F is the fourth chord (IV) and G is the fifth chord (V). Lots of simple songs are based on I,IV and V, so you may recognise that set of chords.