I would have to say( I'm sorry I can't give any rules ;'() that a bullet wound with a little blood like you said would be ok, considering that Rain did get shot, but the only blood was a small red dot-circlly-thing. So, logically, I would say that that is allowed. Also, I agree with the other comments. There doesn't seem to be any problem with a little blood, it is natural after all. When you get cut in a fight(which horses do do a lot) you would bleed. It's true that you may bleed a little more than the artist may choose to protray, but that's understandable.
Hope this helps! Sorry for the lack of rules!

So long as it's not gushing and spurting it should be fine.
I mean, I wrote a fanfic with quite a bit of violence but no bloody descriptions and it was allowed, so I'm sure the inverse is allowed too; some blood but no violence. (By 'violence' I mean you see explicitly what causes the blood, like a really bad fight or a brutal killing... hide the children.)

i'm sure it's fine to send it and really the worst thing that could happen is that the mod just doesn't send it and you can just clean it up a little if it's really bad.

also considering that horses fight allot i would assume yes

I`m pretty sure,a little blood is allowed,cause they allow it on Balto Source,so I would say yes.