Apologies for the bump, but I noticed this question went unanswered...
Kega Fusion has many options regarding controller support, and if I recall correctly, it actually allows you to set the same keys to both controllers - if you haven't changed the player two controller from its default settings, it may be that while you press your own buttons, the game is also pressing player two's buttons and causing him to not only start the game in 2P mode by itself, but also making him punch repeatedly.
What you'll want to do is check the options and make sure that Player 2's controls are either entirely disconnected (should be an option to set Player 2 to "unplugged") or bound to keys that you'll never use on a normal basis (the numeric keypad usually works).
Another oddity I notice, though I don't believe it'll apply to you: if you're using a controller adapter like the Super Joy Box series, and have the adapter connected but there is no controller connected to the adapter, sometimes the adapter will default to having all of its "buttons" pressed and the joystick pressed to one corner. The solution here would be to either have a controller hooked up to it at all times, or just unhook the adapter entirely if you don't have a controller plugged in.
