--"The character running animations plays when you press A but he runs on the spot"
--" the character moves but the animation just stays in one place."
ok aren't those two contradicting statements? First you're saying the animation plays yet the character stays in one spot, then a second later you're saying the character moves, but the animation stays still...
By the looks of your code though, I bet you it's that your character is moving on the x axis fine, but the walking animation of him is stuck. This would be because every time your key down code executes, it's telling your movieclip to go to and play at a certain frame. So the animation never actually gets to play because it's going to the first frame of animation each time. (since your key down code is executing like 30 times per second)
It would be like you trying to count from 1 to 10, but 30 times a second I say "start counting" which makes you start counting from 0, so you never even get to 1.