If a vampire gets a cut or stabbed or just injured in general will it be able to heal itself? And will blood come out? I remember learning last term that to heal yourself you just make new cells?
The odds of this question being closed as "primarily opinion-based" are unbelievably high.
And the only reason I might get away with this is that I can point to Bram Stoker's Dracula and literary analyses to provide some insight.
Vampire stories preceded Bram Stoker's novel by nearly 200 years and show no sign of running out of steam. During those early years the concept of a "vampire" was investigated, sensationalized, abused, and refined. The fundamental concept of a vampire was more than a little solidified by Bram Stoker because (a) he was a whomping good writer and (b) he had the advantage of "modern" printing and distribution (in 1897) which meant a lot of people came to think of vampires the way he did.
In Bram Stoker's novel, a vampire heals faster than normal people and rejuvenates (grows younger) from drinking blood.
If I want to stretch a long, long way, I could suggest that whatever it is that makes a vampire the "living undead" (be it magic, a virus, a curse, or nanites) that "something" is able to extract from living blood what is needed to allow what would otherwise be dead and inoperative cells to do the voodoo that they do to heal.
However, because vampires are utterly fictional, people can and have defined them to be whatever they wished them to be for the sake of their story. Almost inevitably, the creature is sensual (because sex sells), the creature is insanely powerful (be it strength, skills, or abilities... like turning into a
black chicken bat), and the creature has a ridiculous weakness like garlic. I mean, really, what's not to like about garlic?
By the way, that last link will give you amazing insight into the wide variation of vampiric strengths and weaknesses over the centuries. It's very thorough, including even the weakness of the Count Chocula breakfast cereal (the writers have an excellent sense of humor!). I recommend taking a peek!
But, to answer your question: focusing only on Bram Stoker's Dracula: Yes, they will heal. Possibly they will bleed, depending on when they enjoyed their last meal.