I haven't read all the responses, but I'm answereing the original poster.
I would have my horse pts by injection. There is absolutely no need for it to be anything other than stress free for the horse. What should happen, is that the horse would be sedated (and therefore completely unconcious and unaware), before being given a lethal injection to kill it (i.e stop the heart). The down side to this for the horse owner is that the last stage, of the heart stopping, can take some time, then some further time for breathing to stop, together with some "involuntary" muscle movements (which might be construed as the horse feeling something).
But you have to realise that the horse has first of all been sedated (asleep and unaware), and from that point on knows nothing and can feel nothing.
I have been with horses who have been pts by injection by the vet and it was very peaceful. The horse slipped into unconciousness quietly, whilst being stroked by its owner. The owner then chose to leave the stable whilst the final lethal dose was given.
I have no first hand experience of shooting a horse to put it to sleep.
Whilst I favour injection; if my horse was very seriously injured and in a tremendous amount of pain and suffering and moving him either wasn't an option or too traumatic, I would choose a quick death with a gun to put him out of his suffering. Hope I never ever have to do this though.................
I would want the vet to take and dispose of his carcass. Although I would also consider cremation (but hadn't appreciated the costs until I read it on here).