1. DME still doesn't show the correct distance between aircraft and the runaway.
How great is the error? If the error is small, remember that a DME equipment measures the distance between your plane and the VOR itself, like the hypotenuse of a scalene triangle, not the distance on ground.
This generates an inevitable distance error that is bigger the closer you get to the VOR.
2. When the aircraft is pretty close to runaway, the magenta line suddenly disappears and "APP" button turns off. It seems the autopilot leaves the control of the aircraft suddenly. However, the aircraft is not completely in the correct way and it need small changes.
The signal of a VOR disappears right over the VOR itself, as it is an array of directional radio beams. If the signal disappears, maybe is because you are so near the VOR that you find yourself in a "dead zone", so to speak.
3. In landing tutorial "APP" button is always on and magenta line never disappears.
This ties itself to the last part of the previous question. You're following a VOR radial, not an ILS, and it's only normal that the plane it's not perfectly centered with the runway nor the signal behaves like the signal of an ILS. the landing tutorial uses an ILS, that's why you find some differences.
ILS is ILS, Vor is different.

There's but one real cure for human stupidity. It's called DEATH.
At the moment mourning the assassination of sarcasm and irony for the good of the "higher".
Proud FSIX user. Active user of FS98, X-plane and novic