1. Knowing that the horse fell only at a walk and that these supposed injuries usually do not heal 100% can she ever be considered safe for youth to ride ? Being a company safety is very important to us.
One cannot well answer this question without definitive diagnostics of the problem(s). Discomfort due serious pathology seems likely given your description and was certainly a probable cause for the fall. That alone suggests the horse isn't sound for riding, therefore isn't safe.
2. If there is arthritis this usually progresses and since she now has a ligament injury on top of this, can we expect a good prognosis?
Again, without a definitive diagnostic, it is very difficult to provide any kind of prognosis. Arthritis is progressive and doesn't get better. An injury to the proximal suspensory ligament can be serious but many horses do recover. The question becomes, to what level of performance?
3. If she is not going to be safe to ride, can we conisder her for breeding?
It would be unfortunate to ask a horse presenting significant discomfort due arthritis and/or ligament damage to carry the additional weight of an unborn foal. Moreover, with so many well bred, pathology free horses available on the market, why would an organization unable to afford quality diagnostics contemplate the expense of raising a foal?
4. Is the best option a new home where she can be a pasture pet the best option for the horse?
The best option for the horse would be diagnostics that support a remedial protocol of assistance. If that option is beyond budgetary constraints, then yes, finding someone willing to better address the animals needs makes sense. The trouble is in finding someone willing to assume that cost and responsibility.
Sorry I know this is long. I am not a super experienced horse person, I am looking for further answers. We want the best outcome for the program and the horse. Please don't judge.
Regardless the well meaning intent, it's not clear to me that a program unable or unwilling to bear the responsible cost of equine ownership and management, including associated veterinary/farriery care, should be in the business of horses.
Sadly, it may be that the best outcome for the program and that of the horse are incompatible. That is not a judgement; it's a reasonable assessment of the situation as you've provided it.