All sounds normal for a machine that old lmao, fun they are...
I would first remove the starter and see what it looks like inside, for example read this about Fatcats starter, it also has other info that might interest you
http://www.pilotodyssey.com/PO/viewtopic.php?f=1&t=9826 If the starter is bad it should not drain the battery because the only time the starter is connected to the battery is when you push the button to start the Engine, the button is a switch that sends power to another switch called a starter solenoid that sends power from the batter to the starter for as long as the button is pushed, now if the solenoid sticks the starter will continue to turn until the battery is dead. Doubt this is your problem I would first charge the batter and ensure the battery is good, then make sure you have it in neutral or it wont start.
I think some place in the service manual hose routing is shown you might give it a look.
How hot was it outside when you were riding? I ordered a 4" bilge blower for a boat to see if it could be used to force air on a FL350
Engine should be here tomorrow look forward to a review.
Plenty of info on 350 brake systems and all the known problems, rebuilding, bleeding, adjusting etc. any questions just ask we have many brake experts here.
What about the front brakes they working?
Welcome to the site.