If the motor was tilted up when started it is not unreasonable to think the pick up tube will be out of the oil.
I am still confused as to whether the light stayed lit or not.
What light?
Now that I think about it the OP probably had low oil pressure given that hte started with the motor tilted up. Running the motor down probably alleviated the low oil pressure but the light might have remain lit until the motor was turned off. At the next start cycle the light might not be...
No oil level sensor. Only an oil pressure sensor.
You made no mention of an alarm sounding or the motor going into RPM reduction mode. Usually the light, the alarm and the reduction mode all occur at the same time.
Is the light within a Yamaha tachometer or is it a standalone light?
As a side comment, I would say that if the starting battery failed to do its job of cranking the motor I would not parallel (combine) the two batteries without first seeing if the house battery alone would crank the motor on its own. Although perhaps not likely, it has been observed that...
Although they have been out a while, a 9.9 EFI model does not readily compute.
How low in HP will they go?
The amount of gasoline injected at idle must be in the nanograms.
I have seen vibration reducing pads that go between the motor's clamp bracket and the transom.
Switching to a propeller with less pitch can help to slow down the minimum idle trolling speed. As can tilting the motor up somewhat to vector the thrust to a less efficient position.
Piston engines...
With the relay power wires disconnected from the relay connect those wires directly to the battery.
Does the trim pump motor now work or not?
Blue to positive and green to ground the motor will trim up. Green to positive and blue to negative the motor will trim down.
Might be a brief spark...
Yamaha T60 EFI models use a engine control unit (ECU) assembly. No CDI assy.
Stator is not in the loop with respect to ignition. The pulser coil tells the ECU when to terminate the ground to the ignition coil to cause it to discharge electrical power into the spark plug.
Is the ECU receiving...
Not sure I am following you.
If turning the screw outward provides more fuel while turning the screw inward provides less fuel, that equates to it being a fuel/air mixture screw. It varies the fuel/air mixture. But yes, the adjustment is made to achieve best idle characteristics. Fuel/air idle...
No different than anything else that a motor maker specifies.
Ok to use non OEM products "IF" they meet specifications.
"IF" the Relion battery continues to meet Mercury's specifications then it "should" be OK.