Now that I look at it again I don't think it's supposed to be blood exactly, at all, but rather her arteries. They don't look like radial arteries and tributaries to me, but, hey, alien! so that's okay. They seem to be mirrored on each hand.
So, I think what the scene's supposed to be conveying is that her x-ray vision is kicking in. I'd like to point out that there are better ways to portray this (i.e, showing an inset panel of her skeletal structure). But I honestly think this is what they're going for, since her super-hearing starts operating a couple of pages later, and it doesn't look like blood spatter from someone else, and Asrar didn't draw any injury on her.
If they're mirrored then you're probably right.
You know, one thing that building her emergence around a battle scene accomplishes for the character is that she's motivated to discover immediately what her new abilities are. I have to admit that I was paying a lot more attention to her internal monologue than to the details of the fight. And having this as her first experience on Earth is likely to bias her quite a bit.
I'm also curious about her father, and hope that her memories of him have some ongoing relevance to her character and that he was a character in the fullest sense. Her parents in previous versions are pretty much standard-issue Krytonian futuroids, aren't they? I mean, back when I was paying attention (I admit again, it was a while ago) the main way you could tell one Authoritative Kryptonian Male Scientist from another was by the symbol the artist drew on their shirts and whatever the colorist did with their hair. Evidently succession to leadership on the Science Council or whatever is based on a pattern-baldness-and-beard-growing competition that's only open to men of seventy years or older.