Hey, sorry you've had to experience that. I'm also not great with human bodily fluids but fine with all animal stuff (obviously you will have to deal with animals being sick too). I wouldn't say you'll just be surrounded by fainting and throwing up people throughout the course so I wouldn't worry too much, I don't think we've ever actually had anyone do that in my year at all and I'm now in third year so have been through a fair few practicals (although have heard 3 people fainted in 1 practical in the year below). Plus as you go through more practicals people will either get used to it or decide vet school isn't for them and leave, so I think it'll only get better from here. If you've been fine so far I'd assume you have a fair shot of it not suddenly happening to you now, but if you feel queasy feel free to leave the room at least for a bit and have some water, biscuit and a sit down. In the dissection make sure you're not too hot, eat and drink well beforehand and if you feel faint try to focus on shifting the weight on each of your feet for a bit, and if all that fails then do just leave, it's not worth potentially injuring yourself fainting or freaking out vomiting, the lecturers won't mind they've seen it all before. As for dealing with others being sick, not sure how to really help that tbh apart from getting away from the area and trying to distract yourself?
The animals you dissect very much depends on what vet school you go to, at Liverpool we actually don't do many dissections at all compared to what I've heard other vet schools do. Maybe look ahead in your materials/timetable if you can, reach out to a staff member (tutor might be a good shout) or ask someone in the upper years at your vet school. The core species are usually dog, cat, horse, cow, sheep and pig (poultry to lesser extent I guess) but I've heard of other vet schools using the weird and wonderful creatures too so I couldn't say what you'd be using, personally we've only had a full post mortem practical in third year and before that it was a few dissections on specific body parts and instead we usually have prosections so looking at already dissected and preserved specimens.