That "small wire" going from the negative terminal is the ground wire for the ECU. If it's not going back to the computer, that's why it won't start. That wire really needs to be attached to the computer, that's why it goes straight to the battery as opposed to any other ground point. it's kind of important. Go from there... and yes, you probably need a new battery too. Check to make sure both plugs are attached to the alternator, a 3 wire plug near the top and a large wire (B+), black w/ red stripe, that goes onto a stud on the back of the alternator. If they aren't both there, the alternator will not charge. The running rough is probably the Idle Air Controller (IAC). that is reponsible for more rough running at idle than any other part. If it runs rough at all speeds, I would suspect plugs/wires. After that, a fuel pump.