It kind of depends on the geometry of your loader and yours is different from mine so it may vary a bit in operation. Mine is the old style, manually moved stand bars (not the newer, fancier Quik-Tach or whatever they call the complicated setup). If you have the newer complicated style you should probably ignore this.
For mine, curling the bucket to raise the front of the tractor is required in the removal procedure. Why is described in procedure below.
1) Find a flat spot.
2) Raise loader.
3) Lower and lock stand arms.
4) Lower loader to slightly raise front of tractor. This is where the curl is required. If you have the bucket flat or curled back, the stand arms contact the floor/ground before the bucket. They aren’t designed to support the weight of the tractor so if you keep lowering, you’ll bend the support arms. The fix for that is curl the bucket down so it hits the ground before or at the same time as the stand arms so the bucket is taking the weight of the tractor. Just barely raise the front of the tractor. (That’s why you have to curl the bucket and why your tooth bar will have the weight of the front of the tractor on it.)
5) Remove the loader mount pins.
6) Slowly curl the bucket back. The loader should pop out.
7) If it doesn’t come out far enough just curling the bucket back to flat, lower the boom a little. That will actually raise the back of the loader.
8) Remove hydraulic lines.
9) Back away slowly.
So yes, you do have to curl the bucket to raise the front of the tractor with the bucket edge, and yes, you can leave the bucket flat on the floor after it’s off.
Edit: Can the tooth bar take the weight of the front of the tractor? No clue. I take the loader of and put it back on about 4 times/year but don’t have a tooth bar.