I bred my stallion to my own mares first - both stallions were experienced pasture breeders before I purchased them. BUT my own mares were much larger at first, so they had to be caught and handled. Also, not only do I wrap the mares tail - she gets washed with warm water and Ivory liquid and rinsed. If she produces manure (some do!) before she gets bred, she gets washed and rinsed again. Her rump gets washed to both sides of her tail and between her hind legs (basically anywhere he's going to be in contact with her from behind if he "misses"). I don't own breeding hobbles, but do know how to tie up a hind leg to reduce kicking, if necessary. When I started they didn't have mini breeding hobbles and now...? I don't bother. I have twitched mares and used a chain on them as well if needed. I now require that a mare be halter broke, leadable, tieable and I have to be able to touch her (to allow for cleaning, wrapping the tail).
My stallions wear a "special" halter with a stud chain that I run thru their mouths at breeding time (yes, I do work with them with it so they understand it and don't freak out. Freaking out from new equipment sometimes completely destroys the "desire" - one of mine was a shy breeder and if that desire was ended it took forever to "get it back" at first). It doesn't take long, they learn that when that halter comes out, it's time to go courtin'. I also determine a safe place for breeding ahead of time, trying for few distractions, few to no spectators. I try to pick a place that I can "keep" forever as the breeding area. I will tease the mare with a fence between her and the stallion and he also gets washed with warm water and ivory soap and rinsed. I wash the belly in front of the scrotal opening, the whole sheath, the penis and testicles. I do this before every cover but it is especially important between mares.
Now, I love having a heavy wooden fence that the mare can be cross tied in front of. She can't go forwards but can still swing her hindquarters around, so you do have to be careful. Some will kick the stallion handler on purpose! Or I have a set of stocks that I put her in that she can't even move her hind quarters side to side (mine isn't really sized right for the smaller mares - it was built for mares 12-15 hh for my Shetland boys to be able to reach when they stood on a ramp that was also behind the mare when she was in the stocks).
I generally work with the stallion from the left and also the mare from the left - simply because most horses have had the most handling that way and it was easiest. He is introduced to the mare at her head. They can "talk" and strike all they like, though I do try to keep them from striking at each other directly. If you've already teased the mare, she knows what's coming and things will be "OK" if she's ready to breed. He may try to mount from the side, generally striking and pawing if/when he does - I don't yell or hit or yank on him, but tell him "you're not ready" and pull him sideways off of/away from her. I let him smell, sniff, "talk", nibble - he does get corrected for biting or kicking. WE work our way back - some are still ready, some aren't at this point. He has to be fully ready, when he is I say "OK", before I will allow him to mount at her rear (I do let him start slightly from the side). If he shows a tendency to strike or paw at the mare while mounting, he's pulled off and away - my mares were always both ridden or driven and back injuries aren't wanted! Then we try again. Usually the first several "goes" with a youngster were long, drawn out affairs as he learned what was allowed and what wasn't. I REALLY appreciated mares that taught good pasture breeding habits! If he's mounted and "can't seem to find the spot" - I gently grasp him and guide him into the mare.
Being able to handle him at this point helps a lot, LOL. My 2nd stallion wasn't used to this - his eyes would jump out of his head and he'd leap away and Oops, byebye. Might be an hour before I could get him to mount the mare again... I don't know exactly how they felt, but I sure got frustrated with this the first couple of years with him. The first year, even when he didn't "miss", we didn't settle any mares he was bred to. I decided that if we didn't settle mare (s) I bred the 2nd year, that he'd be gelded... Needless to say, one mare did settle. Finally I had mares that were his size and didn't have to help him and he was much happier that way - it's a dance even today with him - 6 years later. Then we settled on a way that worked for both of us - the mares that were his size I washed and covered with him by hand for at least the first cover. Then they get turned out into his pasture with him - easier for both of us. With a larger mare or with an outside mare (I haven't had any for him), we'd do all hand covers.
Contract - I have a very basic one that includes mare care and stud fee blanks that I filled in. I didn't used to have insurance, now - I would have to think about that... I had a $50 non-refundable booking fee and a $200 stud fee (at the time). Most of the mares coming to us used the same hay we did, they provided their feed while the mare was at our place and I fed her. The mare stayed at least thru the heat cycle, several stayed longer than that. I DID want at vet proof of a negative culture (that doesn't always show if a mare is clean or can support a fetus thru a full pregnancy, just means she's clean of infection drainage that will affect your boy), negative Coggins, Rabies and EWT. If they had received other shots, I wanted to know when (strangles then was a live vaccine and I didn't want a mare newly vaxxed w/ it on my property since I didn't give it). Now, I also require WN. I don't necessarily state that they have to have Flu/Rhino (most did) - now, I probably will... Plus will be keeping them away from at least my babies and mares. Next year I have no foals coming - so not an issue.
Next year, will do $100 non-refundable booking fee & $250 on a jr stallion that isn't proven yet - have two mares lined up for him. Will be breeding to a couple of my mares first to teach him and then to make sure not doing "blanks". He has been shown in halter (doesn't have his HOF yet), has awesome movement for what we want and is starting driving training now. I'm actually pretty excited right now over him. He'll be 3 next year. He developed study characteristics and lots of screaming "idiotness" while with a show trainer. He was a complete and blithering "idiot" when I brought him home in July. He's pastured with a bunch of geldings and other junior stallions and has junior fillies in the pasture next to them. Not the best scenario - it has been a bit problematic - the other boys have taken a bit of a beating, but he's also learned to get along with them finally and now, some of them go after him as well. This may change next spring when the mares start coming into season! We have started a proper stud pen for him and will have to have it finished, but for as long as possible, I want to leave him out with the boys. I just feel that it's more healthy for everyone.
My first stallion learned rather quickly that he had two jobs - one was working: either driving or being ridden. Two was running with his mares & breeding. When he wasn't wearing his "breeding" halter or turned out w/ the mares - he was required to "be quiet". Once he learned this - our daughter often trail rode him w/o anyone even realizing that he was a stallion for several years (to this day she still asks where she got the 6' height gene from!). He was even utilized in a beginner rider program (but he was on a lead or lounge line and by himself in the arena then). We did take him to various events with other horses and he wasn't a problem. Kids loved him when we used him as a lead line pony for day cares, churches and Halloween events. When our middle daughter took lessons on him at a local stable - he was ridden in the arena with other ponies/horses (from our place), but not with other horses of any sex that he didn't already know. Our current stallion hasn't completely learned this yet - this year, maybe because we haven't bred any mares, he wants to talk to anyone anytime he's led out of his pasture - breeding halter or not. He gets corrected for that. He stops. He's been pair driving with both a mare (she hasn't been in season yet when hooked w/ him, that may be interesting) and his 3 yr old gelded son. He's been better behaved and quieter than his son!!
At shows, you will need to correct him for talking and for dropping (I use "no" and "EH EH" for talking and tap his penis firmly with a crop, whip or lead line and say "put it away" or "away"). They learn. You do have to watch for striking hooves and other "careless handlers"... I don't want someone allowing their horse of any sex getting close to my boys and inviting problems (actually I don't like anyone letting horses get into another horses' space. I'm really bothered by what I see at some of the shows I've attended the last several years).
The only thing I know about breeding donkeys and horses is this - if it's a female donkey (jenny) to a male horse (stallion), it's not a mule - the result is called a hinny. It is more horse like than a mule will be - both the noise they make (more like a whinny?), and their manes and tails and body appearance. If it's a female horse (mare) to a male donkey (jack) - it's a mule and is more donkey like in appearance and sound (brays). Both sexes of either hybrid is generally sterile but males are still castrated/gelded. Some have become pregnant or sired offspring. I don't know if the hinny has the same personality traits that a mule has, but I know that I'm not a mule person, LOL.