Further to what others have said... Check out Equine Cushings & Insulin Resistance Group
for GOOD info on the subject/diseases.
If your horse is fat, she is not *necessarily* EMS or such. Cushings is not necessarily a dietary disease, but is due to chronic stress on the body(commonly but not always laminitis & obesity) which causes inflammation, particularly in the pituitary gland. If your horse has what you describe as 'fat pads' it is, however, very likely(without seeing, not sure...) she is obese, not just 'fat', and it has been a chronic problem. If 'fat pads' are hard, it IS likely she is EMS/IR.
DRY hay is not necessarily any different, in regards to sugar content, as fresh grass. Sugars are only used/lost in active growing, so once grass is cut, sugars are 'locked in', unless the hay is soaked & drained before feeding, to leach out sugars. I'm not sure about Timothy, whether it's particularly rich, and with any grass, it depends how it's grown & when it's cut as to how much sugar, but if you're not sure, you can soak a little of it, in a little water for a while, and the browner the water that comes off it, the higher sugar content - then you have a better idea of whether you should leach it before feeding or not, to 'at risk' horses.
If your little pony is obese, there's a fair bet your mule is too, as with her donkey blood, she's likely an extremely easy keeper, more at risk of EMS/diet related laminitis than a horse that size.
Re how much time/when to graze, remember grasses produce/increase sugars with photosynthesis, and use them over night in growing. So the safest time to graze *generally* is in the 'wee hours' and early morning, with highest sugar content being in the afternoon/early evening of a sunny day. BUT grasses 'shut down' & don't grow when stressed, including when there's frost, so can continue to just accumulate sugars without using them up. The frost factor is biggest in spring & autumn, which is why I think they are the riskiest times of year for laminitis.