1 - When I started my bike this morning, I noticed it was making a slow thumping noise at idle and I could just feel it. Never noticed it before and I am one of those that listen out for all noises all the time! The noise disappeared when the bike was warm. Trying to figure out what it is. It's not a rattle it's a thump.

2 - I am due for a tire change and I got a reminder today when I got the K sideways big time coming up to a junction, did manage to keep it all upright though. My fault, had my head down listening for the thumping noise, looked up and "Oh Sh!t!"................. Like I said, my fault.

3 - When I reached my destination, I took a look at the rear tire to see how low the thread was when I noticed a drip mark on the tire, that must have happened when the tire was stationary. Looked at the final drive drain plug and forming there was the start of a droplet of oil. First time I have noticed it, but I know the drive was not dripping as the floor has no oil stains on it, and there are no other drip marks on the tire. Thought the plug was loose but when I tried to tighten it, it was nice and tight. Have wiped the FD so I can check it later to see where the leak is coming from.

It may not be that unusual. The noise. Especially if it was colder out doors this am?
Most early K's had a built in factory defect in the design of the engine output shaft gear tensioning device. I'm just guessing here but the output shaft driven gear (driven from the crankshaft drive gear) has some lash between the two gears. The "slack" is supposed to be taken up by a "spring loaded" (for want of a better word) wafer gear adjacent to the driven gear that has a circular annular spring & mounting pin mechanism that is supposed to put some "torsional load" on those gear teeth to keep the lash between them from knocking. The spring design was poor and most of the early bikes have no tension left on the gears from that spring. ....Rattles quite badly....the colder it is, the worse it gets. Mine sounds horrible when first started up at 0 deg. C.
Another symptom sounds like a Growling noise that many mistake for a timing chain problem (again while cold) on rapid deceleration.
As soon as the engine gets fully warmed, most of the objectionable noise dissipates. There is a BMW bulletin around here somewhere on how to fix it but the whole bottom half of the engine has to come apart including pulling out the transmission to accomplish the fix.
Additionally, in some cases, another part of that same output shaft can develop some loose rivets heading up to a subsequent failure. The 2 faults are not directly related to each other. The fix for the rivet problem if it occurs is to install an updated shaft with 12 rivets instead of 6.
I understand your tire issue......I'm not sure what to make of your oily mystery. I'm wondering if you just got in over your head when it went sideways??????
Or perhaps some of the oily mystery stuff is flowing out onto the rear tire?

+1 fot that ibj I am thinking the oil will be coming from the seal on the crown gear between the disc and rear drive case.
It usually doesn't leak enough to give hassles on the tyre but can get on the disc and make the rear brake even more ineffective than it already is, if that is possible.

Better get the bike up on the center stand and do the 12 + 6 and the 9 + 3 (o'clock) wiggle test on the rear wheel

Yes it's been getting colder but no where near zero! So don't worry about the thumping for now? Just monitor?

Will check the rear wheel again tomorrow and the crown seal. The drip wasn't much, in fact it looked like one drop that slowly ran down the side of the tire, there was very little.

The clue is that the noise goes almost fully away when warmed up.
Something else I forgot to mention, If it dissipates or disappears when you dis-engage the clutch it could be normal sounds caused by the trans in N.

Mines a bit similar when cold, I thought it might be the sprag starter clutch catching. I would change the gearbox oil if required, I find new engine oil at target top circle level seems to help a bit re gear noise?? not sure why. when the engine is running the level drops due to circulation volume.
Rick

Ok no more drips and rear wheel passed the wiggle test.

Changed gearbox and FD oils today and the gearbox is much smoother. Used Syntrax 77/90W in the gearbox. Used a dino oil for the FD though, as I read on the portal that using syntetic there might results in leaks.

The thumping does go away when the engine is warm so I guess it's ok. Will continue to monitor.

Thanks guys!

I wouldn't go so far as to say that the noise that goes away when warm is "ok", but given the amount of labor and money required to go all the way in to correct it, it's probably an acceptable risk.
If I were going inside to repair the shaft I would think that would also be the time to replace the long overdue timing chain, rails & tensioner.
For me, those two repairs alone in time & materials cost, would exceed the value of the vehicle.