Mares do have hymens, but if this was the third cycle she's been bred chances are anything left of her hymen is completely gone.
Largely endowed stallions can certainly cause bruising when they thrust or tearing when they flare. Breeding rolls, mentioned above, can help with the bruising part. They are basically a small roll of paper towels attached to a pole. The roll goes between the stallion and the mare, somewhere around the level of the mare's anus, and prevents him from entering her too far.
It is very possible, but rare, to have a stallion actually rupture her vagina and ejaculate inside her body cavity completely outside her reproductive tract. Strangely enough this is not usually life threatening... unless there is a major vein/artery involved the mare will heal very quickly. If any blood is seen, a speculum should be carefully inserted to examine the vagina for tears or bruises. This isn't something you should do if you've never done it before... the risk with a normal mare is almost nil, but with an injured mare you can make things worse if you aren't careful. This is a skill that every breeder should practice at some time during their career. Disposable speculums cost only a few dollars. Your vet should be able to teach you pretty easily.
Don't forget that just because the blood is coming out of her doesn't mean its her blood. Check the stallion as well. Make sure his penis is uninjured. Check his urethra for signs of blood. It could have been caused by an errent tail hair (hence why tail wrapping is so important), from debris on her vulva or his penis (hence why washing them both is important), or from an internal problem with him. If he has blood in his semen it can indicate minor irritation of his urethra or something as serious as an internal bleed/infection.
Blood of any kind is not supposed to be outside of their body's... so if you see it definately act like a detective and track the source down. If you aren't comfortable with this you can always call your veterinarian to come help. Luckily, most of the troubleshooting you can do yourself with very few tools, and if you have to call the vet you can usually tell them at least what's happening and let them dig deeper into the problem if its something you can't help.
Luckily the reproductive system is quite strong and resilient, and because of its high blood flow and little muscle tissue (in the mare) things heal very quickly. Stallion issues are usually more serious, and can lead to infertility or penile amputation in extreme cases. But luckily, most of the time the blood is just from minor cuts or irritation on either of the two partners, and a few days of rest is all that's needed to make things good as new.