You have no fatal injuries, and if you did, you'd probably pass out at some point. Since that's the case, and you're in the Order HQ, someone would take care of you if it came to that.
It's doubtful the Order HQ will be attacked any time soon, so there's no IMMEDIATE need to get patched up.
However, what we know right now is that an attempt on your soul has already taken place, as evidenced by the protection mark thing getting burning hot. This means that any kind of demon stuff could be happening and you might not know anything about it.
To further push this point, multiple possessions of yours have been around demons a little too much for comfort. Lets point them out:
Knife: Covered in demon goo. Who knows what will happen to this thing. Revolver: Was OWNED by a demon. Who knows what's gone on with this thing. Pistol: Was knocked away by a demon. Admittedly nothing probably happened to this thing, but it's best to check anyways. Shotgun: You dropped this yourself, but since it wasn't in your hands at all times, might as well get it checked.
Further on, since any of these could be harboring something negative, and you currently posses them, you yourself could be in deep trouble, possibly, just for carrying them. It could lead to dire consequences like wounds that won't heal. And it's best to minimize the risk to others by seeking immediate treatment.
This is also why, even if the Revolver is Order-owned, it should be checked out first. Don't want it sitting locked up in case while harboring evil and spreading whatever the heck it might be able to spread.
However, wounds to yourself are only a risk to you, and it takes a very long time to die of blood loss, and you haven't lost that much and aren't seriously bleeding out. Though even if you were, there's plenty of people to help get you medical attention.
TO TOP IT ALL OFF:
Vergil and the rest of the Order probably think about this kind of stuff too, so this is likely also part of the rules if it actually really is a serious concern. So Vergil might get mad if you try to go and do some other stuff before anything else.
One for all, right? In this case, keep the Order into account before you go make yourself feel better. And then after that ensure your medical safety.
It's like my Rattata is in the top percentage of Rattatas.
While being limited to one choice is challenging, the choices presented are definitely viable to be grouped together.
Getting check for demonic residue is a priority, because even if the sigil prevented them from getting possessed by that demon they killed, who knows if its blood could somehow bypass the sigil and do some nasty **** from direct contact with pre-existing wounds. I say get checked for any demonic residue and stuff (also making sure that the sigil is still intact and will function as it should, and possibly get it slightly upgraded somehow), then get your face down to the infirmary and get patched up (no need to go reporting to vergil looking like ****), THEN report to vergil with your findings
We can discuss what to do with Bosch's revolver later after it too has been checked for demonic taint
"I absolve you of all of your sins, now get the **** out"<-----------Machete Xbox live GT: King Kickass99