Does a Pokemon's Attack stat affect the Pokemon's move damage?

Solution 1:

Yes. The Attack stat affects the moves in addition to the non-move normal attacks. If you use a move from a low level Pokemon, and then use the same move from a much higher level Pokemon, you will notice a substantial difference in their damage output.

It is important that the moves deal more damage with a greater Attack stat, since with a set damage output, otherwise the move would deal too much damage when Pokemon are at low levels, too little damage when Pokemon are at high levels, or both (in a similar vein to the move Dragon Rage in the mainstream Pokemon games which always deals 40 damage).