Unlevel Toto Drake!
I just finished making our full bath half functional (shower is not done). The floor is tiled with large format tile and verified level with less 1/16 lipage. The flange is new and is also level to the floor. When I installed my brand new Toto Drake toilet I had to shim it 1/4 of an inch in the front to get the darn thing level! And then when I put the tank lid on it either sits ok (not great) looking in the front or the back and if you push from the side it will slide off! I'm going to have to use putty to keep it on as this is a childs bathroom! This is not my 1st toilet installation and I have never had these problems before used American Standards before. Any one else had these problems with a Toto Drake (round)? Did I get a lemon? Sorry I'm venting a little I expected more from this toilet!
On a positive note it flushes like a dream.
I would suggest you go back to square one and start over. Remove the toilet, clean off the wax ring from both the toilet and the flange. Then set the toilet in place without a wax ring. From you description of the floor, the toilet should sit level on the floor. If it does not, inspect the base of the toilet for a casting flaw. Use a straight edge on the base to find any high spots. Obviously, there should be none, but something appears to be preventing the toilet for setting properly. If you determine there is a flaw, work with the dealer to exchange for a new toilet. Toto has an excellent reputation for quality, but perhaps 1% slip by with a problem. With a level floor, level flange, and smooth toilet base, no shimming should be needed. If this suggestion doesn't work, report back and we can put our collective heads together.
All toilets are made of clay; formed wet and then dried in a kiln. They lose about 25% to 30% of their size in the process.
The only thing we care about is installing with the front of the bowl down on the floor, shimming the bowl in back if the floor is uneven.
I would rather have the bowl lean forward and keep more water in the bowl then the other way around, with less water in the bowl.
The lid should sit on the tank just fine by it's own weight. I don't have any customers with kids calling me to say that the kids were playing with the tank and broke them. But I'll let you know if that becomes a fad with the kids. So far, it's those little tablets that they play games on.