They are in VERY poor shape. The reason her sole is bulging out like that is because she has so much retained sole.
You and greenhaven are both incorrect in this case. There is every indication that there is NO false sole and in act the sole that is present may be is prolapsed. It is a FOUNDERED foot where the bone has dropped and pushed the sole down with it. Lateral view X-rays would conform this.
The KEY to telling whether there is false sole or NOT is the depth around the frog apex. In this case, it is NOT deep.
That is the TRUE sole we are looking at and if someone decided they would try to remove it, thinking this was false sole they may just kill this donk.
The laminae in the dorsal hoof wall has been seriously compromised and the repeated abscessing may just be dead torn tissues turning to serum and finding a way out. If the leakage from the abscesses is clearish, reddish brown and doesn't smell as bad as pus, it is probably serum, it is extremely common in chronic founder. Unfortunately everything that could help the most would require the cooperation of the donkey for the farrier. So the best you can probably do at this point is keep the heels trimmed back so the frog can help support the bone, and try to keep rasping at the front hoof wall to keep leverage off the toe .
And of course really look at thw total diet and remove any hidden sources of carbohydrates, Purina products in general tend to be way higher in carbs than they should so I would switch to a guaranteed low starch feed , WEIGH the grass hay to limit its intake to minimum maintanance ,and maybe try adding a magnesium chromium supplement like Quiessence, and some cinnamon. Dietary changes and supplements designed for metabolic issues are a "cant hurt" approach and could help a lot. .