Ok, now that part is answered
Tiring quickly could be due to a few different types of illnesses
The fact the horse is over weight and tires quickly may indicate it has insulin issues and it's also possible the horse foundered somewhere in the past.
Either way, it explains why the original owner may have wanted it gone and felt the burden of guilt lifted off their shoulders by saying if someone didn't take the horse for free, they would put it to sleep
Or, on the up side, it could be the horse isn't as bad as the original owner thought and this could be just matter of putting the horse on a diet, before it does get sick
Can the current owner afford to have the vet look at the horse? That would be the first and best place to start. Making false assumptions as to why the horse is lethargic could cause more harm.
Does anybody have any idea how old the horse is?
Are you in the United States? Where you live is going to make a difference on feed recommendations
Limited pasture time or put a grazing muzzle on the horse during daylight hours is a good place to start, regardless of where one lives. Sugar abounds in grass all over the world:)
As far as exercise, if the horse can tolerate some hand walking, that's all I would do and keep her in the soft dirt, grass, fields. A tractor lane would be ok, as long as it's not full of rocks.
If the horse is dealing with some sort of metabolic issues and hasn't foundered too horribly in the past, it is possible to make it's life more comfortable but probably not to the point where it would ever be more useable than a short easy hack or to put a child on its back for a few minutes.
Is it possible to get some clear pictures of the horse from the side and also the hooves? Outside during the day are the best