well, I've been showing horses since 1976--some in breed shows, a lot in open shows and fairs, in halter, pleasure, driving, dressage...I haven't showed dogs and have never even watched a dog show so cannot say how showing horses compares to showing dogs.
I have had people try to cut me off in pleasure classes. In some cases I wasn't sure if it was a deliberate attempt to sabotage my ride, or if it was just stupidity on the part of the other rider. I have watched classes where someone got cut off, and it was quite obviously a deliberate group effort to ruin the ride of a specific person. I can think of one instance where a particular rider was being a royal pain to other riders--she wasn't deliberately trying to get in her way but in her arrogance she was doing her own thing with no consideration for the others in the class and, well, lets just say that some of the others got even!
I have been ring steward at a breed show where an exhibitor was being very pushy to the others. I'm convinced that he wasn't deliberately trying to ruin anyone else's showing, but he was so determined to get across to the judge "me, me, look at ME!" that he totally didn't care that he was being pushy to others. Before it got to the point where he was actually ruining the position of other horses and the other exhibitors started rolling their eyes and sending ferocious glares his way I was telling him to move up here, and please move NOW, and gave him my own best glare to make the point that I meant business
Definitely you have to look out for yourself. If you're in a performance class, always be aware of where the other horses are and do your best to keep yourself in a good position. Don't let yourself get boxed in. If you're driving on the rail and catching up to a slower horse, and you're aware that there's another horse coming up behind you, pull out to go around the slower horse a little sooner, rather than leaving it a little later. If you wait a little too long the other horse may come up beside you and then if he should slow down--for any reason, whether it is deliberate, or just for some reason he does slow down--you can find yourself stuck on the rail with no option except to break down to a walk to avoid running up on another horse. If the rail is clear ahead of you and someone is passing, just be aware of that, and "ready" so that if the other horse suddenly cuts in front of you, you are ready to deal with that.
Halter classes--again, just watch out for other horses. One might be unruly, or the handler might be careless and lead their horse too close--I can't say that I've ever seen any halter handlers try to mess up the showing for other competitors, but I have seen careless moves cause problems for another competitor. You should always be aware of where the other exhibitors are around you, but when it comes to halter classes I have to say that I think it's extremely unlikely that anyone is going to try and 'mess you up'
In horse showing you will meet some very friendly people, and yes, you will also meet up with some unfriendly ones. There are some real sourpusses out there! I was at a show on Saturday and nearly everyone was really nice & friendly & pleasant, but there were two that would surely have felt pain if they had smiled or acted pleasant toward anyone! I ignore that kind and enjoy the rest!!