Folks do this in a lot of different ways, but here are some ideas from the ways some well established facilities around here do it...
Say I have two people scheduled to come out a few days apart. If the first person were to say they were interested and plan to get a vet check that would not take place until after the second visit, would it be best to cancel the visit with the next person and put them on hold? Or have them come out and inform them that somebody has a check scheduled and may buy? Or? I would think you would cancel their meeting for now but I want to be sure.
The vet check is the last step after the buyer has decided they want the horse, so the seller will require a deposit to 'hold' the horse pending the vet check and the buyer has so many days to arrange the vet check.
This includes a committment that if the vet check is OK, the buyer will follow through on the purchase. If they do not follow through (e.g. just change their mind), they forfeit their deposit. If there are any problems with the vet check, the buyer has the option to continue the purchase or have their deposit returned.
During this 'holding' period, the seller cannot sell the horse to anyone else, though the seller can still show it to others if they are informed that the horse is pending sale. If appointments are already scheduled, you should call to inform them of the pending sale and give them the option to cancel, reschedule, or keep the appointment.
I do plan to have a selling contract to be safe. I will probably select one of the ones available online. If I want to allow a trial period, should I have another contract made up stating something along the lines that the potential buyer is responsible for any injury not directly related to any pre-existing condition or anything like that that takes place during the trial? Or do we have to risk it? And if there is a contract involved, is there anything else I should include?
Most places I know are reluctant to accept trial periods (especially if the horse is to be moved) because of the potential risks. Be very carefully if you go this route. I'm sure others can give you advice about trials.
Based on things I've read, I'm thinking that I will arrive at the stable a bit before they arrive to get him cleaned up and looking his best, then put him back in his stall until they arrive, then retrieve him from his stall again while they are present. Is that correct? I read that it's best to have the grooming supplies and tack out by the tying area but have the horse in his stall so that they can see how he is getting caught. From there, if he is kept in his stall, should I offer them to get him out? If I get him out, should I lead him to the tying area or offer them to? I really don't know what is customary.
Typically, as you mentioned, the seller will have the horse cleaned up, stalled, and ready to show before the seller arrives unless the buyer has requested something else. Make sure you understand the buyer's experience and skill before letting the buyer handle the horse, and if at any time you feel uncomfortable with the buyer, do not hesitate to end the appointment.