They're probably not related. The intermittent running usually means that the flapper valve is leaking...the sound of water running is just that, after the water leaks far enough, the valve turns on to refill the tank. Shut the supply off and leave it...my guess is that the tank will be empty after awhile. Replace the flapper, or make sure there's nothing caught on the sealing surface preventing it from making good contact.
The slow flush is one of two things: there's something caught in the toilet trap (pen, hair pin, toothbrush, toy, etc.) or the main line is partially clogged. WHen it's partially clogged, it depends on how recently you've run water somewhere and how far the pipe has filled up. Since it's not a complete block, after awhile, it drains, giving you room to flush again. If it isn't in the toilet, it's usually time to call a pro.