Actually blood in the stool doesn't necessarily = cocci, particularly in 'adult' hens and ones about to lay (around 6 months) but it's certainly something to rule out (eggdicted - more for you in a separate post from me in this thread). It can also be bacterial irritation of the cecum, worms, egg binding, or (possibly in this case) blood from the vent protrusion.
Would you say that the vent looks like this:
If you feel this is it, then here's an article I think you will find useful. Note: This is the ONLY time I would consider using any 'caine on the bird at all, and the bird must in all cases be separated from the others.
http://www.browneggblueegg.com/Article/ProlapsedOviduct/ProlapsedOviduct.html
Or does it look more like a sore at the side of the vent.
Also, has the hen laid an egg yet? If you very delicately feel the abdomen, do you feel what could be an egg inside? Is it possible that any of that dropping was actual blood from trying to lay an egg? Was the stool mucousy?
If the stool was mucousy and the blood is in that mucus, then I'd consider coccidiosis. But it's not really very high on my list. I'm thinking more wound or prolapse based on your answer. I'm leaning more towards prolapse as you used the term hemorrhoid and as she's about the age of beginning laying. Beginning layers can often push too hard and prolapse their cloaca, particularly if they have a soft shelled egg, etc.
In case of a wound only, you will want to cleanse the area (preferably with a mixture of betadine iodine with warm water til it's the color of a very weak iced tea). Pat dry once it's quite thoroughly washed. Dress with neosporin (with no pain killers and no cortisones).
In ALL cases keep this hen inside and separated from other birds. Birds will pick her til she's dead. Flies can easily get to a wound/prolapse like that and will lay eggs, causing maggots within hours in the summer. So she must be inside while you treat as long as there's anything red and raw outside.
We'll look for your answers so we can figure out whether to go more towards prolapse or wounds. In the mean time, consider vitamins/electrolytes (for poultry use as labeled) in the water, make sure she eats and drinks, and clean off the area at least with warm water.