582 leaking oil (!?) from exhaust

I am also curious to hear more. I have just expirenced the same. Fitted the Oil injection (the pump was there but never been used) followed the procedure from the video in Rotax-Owner.com and added 1% Oil to the gasoline as recommended by Rotax. The startup went Well but Black Oil Seem to be "thrown" out of the exhaust and after engine shut Down the Oil was dripping from the exhaust. I do not have egt readings. But the water temp was fine 

You are burning a fuel/oil mix, incomplete combustion results in oil being visible in the exhaust. A good tuneup will eliminate most of it but it is the nature of the beast. 

Remove your air cleaner and see if you have a build-up of injection oil sitting in the carb inlet. Also, remove your spark plugs and see if they are wet/black/oily. The little check valves in your oil injection pump could be leaking allowing the injection oil to build up in your carb. IF they are leaking bad enough, injection oil could be sitting at the bottom of your crankcase. Once you start the engine, this injection oil at the bottom of the case will get into your exhaust... I suspect the reason for the power loss is your plugs are oil fouled. 

(I stand corrected! YOU can change these check valves!!) 

Took off the air cleaner.  Both carbs are shiny clean.  A little oil on the carb slider on the #2 cylinder (Mag end), but it's the #1 (PTO end) EGT probe that's dripping oil.

Ran the engine for a minute or so, and shut down when oil started dripping from the exhaust ball joint.

Pulled the plugs, all are wet.  #2 plugs look like mostly gas, #1 plugs are also oily.

Where are those check valves, and how hard are they to swap?

Lift the piston slides in the carbs and look behind them. The injection tube sits right behind the piston slides and is where you would see a build-up of oil. (I apologize for saying they are in the inlet when they are behind the piston slides, brian not working properly)

In the attachment is the check valves from the Rotax IPC, which can be downloaded from here - https://www.rotax-owner.com/en/support-topmenu/engine-manuals#582-engines

Section 79-00-00 Page 152

Item# 26,25,24,22 

They are simple to change!