Answer:
No, rabbits don't menstruate or bleed when they're in heat. There's no healthy reason for a rabbit to bleed! If a rabbit is bleeding from its genitals, it could have an infection, or an injury, or perhaps a tumour, or maybe something else -- the point is: something is very wrong! If your rabbit is bleeding from the genitals, it needs attention from a vet with experience and knowledge in rabbits as soon as possible.
Rabbit urine can sometimes look red for healthy reasons, and it doesn't mean there's any blood. The vet can tell you whether your rabbit's urine is bloody or not. Another way to tell is to put an old white towel where the rabbit pees: after it pees on the towel, look at it closely. If the red colour is all over where the wetness is, that's not blood, it's just pee; but if the redness occurs as blotches in the wetness, and some of the wetness isn't read at all, that could indicate blood and a serious problem!
See the related questions below for more help and useful links.