Will a Toto Eco-Drake toilet with a 12“ rough-in fit the 10.75” rough-in

It seems as if you have an obstruction that may be floating around in the pipe or valve. Sometimes it obstructs sometimes it flows clear. I would start with changing the water valve. If this is the old style "gate" valve there are several things that can clog the line. I would replace it with a 1/4 turn ball valve. That will give you much better flow than a gate valve. You will need to shut down the water probably at the main. The old valve is probably soldered, but you could cut it off and use a compression type valve instead. "Sharkbite" is one such supplier. Although I have never used their compression valve, I have used their end caps and found them to work well.

If you can have someone hold a bucket under the pipe once you cut off the valve and then a second person turns on the water main to fill the bucket to purge the line of any debris. With cast iron pipes it is common to get clogs from rust build up. The rust may not pass the small opening in the old gate valve when you first purged the line.. Although you did not mention what type of pipes you have, if it's copper than you won't normally find rust obstructions..

"If left alone with water input off, the tank water level will lower (perhaps 1/4" in 1 or 2 hours), and puddle will be present near front of toilet. This suggests to me a bad job of installing wax seal"

There is an important clue buried within your question. There is no water from the toilet present at the wax-ring seal except when flushing.

Your problem/leak had nothing to do with your wax ring!