I assume you are talking about the starter solenoid: Follow the positive battery cable from the battery - the other end will be connected to the starter solenoid.
under the seat, connected to battery
The starter solenoid should be located on the inside fender close to the battery. It will have the positive battery cable connected to it.
The wire that goes to the starter button can be connected to the power wire coming from the battery. Just tough the wire quickly. If the solenoid engages the starter, then it is ok. If not, then there is either no power from the battery of the solenoid is bad.
The starter needs a large wire from the battery and a wire from the starter switch. These are connected to the solenoid that is attached to the starter.
The starter solenoid is attached to the starter. Follow the positive battery cable from the battery, it will lead you to the starter solenoid. The starter would have to be removed to replace the starter solenoid.
Yes, a starter solenoid can drain a battery. When a starter solenoid is not working properly it can actually keep rotating the starter after the car's ignition is turned off. Since it requires a battery to actuate the starter, the battery will eventually be drained of power.
I am not completely sure on why you battery in your Mustang would be smoking only when connected to the starter. I would take into a shop and have them look at it.
Ford's fusible links are usually connected to the battery side of the starter solenoid.
The starter solenoid is mounted directly to the starter. Just follow the positive battery cable from the battery, it will lead you right to the starter solenoid.
starter solenoid is connected to top of starter
Your problem is the starter solenoid. It may be a remote solenoid mounted on the inner fender well. You can easily check it by running a battery booster cable from the positive battery post to the starter side of the solenoid wiring (the heavy wiring). If the starter operates, the solenoid is bad.