I also, have had several horses put down here over the years. The first was my own little WB mare. She was the first horse I had bought with my own money when I was a kid, loved her to bits. I got home from school one day and usually she would come galloping up to the gate the greet me, this time she neighed in a very high pitched panicked tone.
Turns out she had broken her front leg all the way through the cannon bone, to the point where the leg was dangling by a bit of skin. No one else was home, so I had to leave her to call the vet, and then stand with her for almost an hour until they could get there.
She was so stressed that the vet could not get near her to inject her, so shooting was the only option. She was gone in an instant.
We were able to get a backhoe in and bury her on the property the next day.
Another horse, was my coaches 30 year old gelding. He was going downhill fast, full of cancer and arthritis so the decision was made to put him down. He was weak and sore, so ready to go. He was given an injection... it was one of the most traumatic things I've witnessed. He fought the drug till the end. Even when he couldn't stand anymore, he was on the ground just about doing cartwheels. He was thrashing so violently that the vet couldn't get near him to give him a second dose and so he died slowly and violently.
The others again were retired horses put on my property by their owners. All of them were shot. I could not cope with seeing another chemical euthanisia. I know they're not all like that, in fact what I saw would have been quite rare, for a horse to fight to that extent, but it is really traumatised me and now it is shoot or nothing. Shooting is immediate, the horse has absolutely no idea of whats coming, no pain, no anything. One second it's happily there munching on some hay, and the next, lights are out. I know shooting sounds awful, and in this day and age we're brought up to think that shooting is terribly inhumane, but actually, it is kinder than chemical. There is absolutely no chance of the horse fighting it.
As for getting the body removed. Don't be present. As someone above said, it's not a nice process. One of the retirees was put down on my driveway for ease of access of the truck. There was a blood stain for weeks from him being dragged up the drive by chains into the back of the truck.
I'm really sorry, this is probably making you feel awful!! I don't mean it to, once a horse is gone, it's gone and can't feel a thing. I think we as humans get so upset about these things not so much because we're worried about the horse, but because we've been left behind and have that feeling of guilt that we justified taking a life.
Remember that its' the right thing to do for the horse, if he's sore and losing weight, it's far kinder to put him to sleep than let him go on for years. The unfortunate thing about horses, is that they don't tend to 'just die one day' and rather, just slowly degrade until they're skin and bones and end up colicking :(
You have made the right decision, stay strong and big hugs to your mum.