Sorry to disagree with the tall guy, but the amount of storage space used has nothing to do with performance UNLESS you fill the device to the point it runs out of room for writing temp files, cache files, etc. At that point it will become slow and unstable, just as Windows PC does when the HD is nearly full. But up to the point it runs out of space, performance is not affected.
A factory reset will only speed up the phone if it has a problem. Something like some software misbehaving or a major system upgrade that didn't go smoothly. The first usually will not require a reset to correct. The second might be fixed with a cache wipe instead if a full reset. But if everything is operating normally a factory reset will have no effect on performance.
So... No, downloading and playing games will not by itself slow down your phone.
Android since v1.0. Linux since 2001