Battery slowly discharges on my 2002 FXD when engine runns

This is a very interesting situation. It seems like you have a situation that when it is cold - some contact stay open and when it gets hot = the contact closes and drain your battery.
Do you have an automotive scanner to test for faults?
This is where I would start as it could be anything from your car computer to something very simple.
I would replace your thermostat. and check the sensor at the thermostat. Using a test light and connecting it to the sensor - it should not light up as the connection is not connected. The connection inside the sensor should ground out when it is over heating. or just replace the sensor to see what happens.

Don't know.
You would need to have both components tested. It could be a dead cell in the battery. Or a bad connection. If the alternator were at fault it should discharge when the engine is running and slowly go dead.

too many variables !
Firstly check battery under load ! any battery place or auto electrician can so it usually free!
do this before replacing any thing else!
make sure all connections are clean & tight
if battery is ok get them to check the charge rate on bike! its a simple test or you can do it with a volt meter they are only cheap to buy!

Loose wire connections must be checked first.
Here is the simple way to check if the battery is discharged: Turn the ignition on, with the engine off, switch the windshield wipers on. If they move way too slowly, much slower than usual, the battery is probably discharged.
As the battery is relatively new, it can be recharged. All you need to do is to boost your car and let the engine run for a while to charge the battery