Can Superman lacerate himself? If so, could he die from massive blood loss?
As to whether he can lacerate himself, the answer is a resounding yes;
As to whether blood loss will kill him, again we can see that Superman and his doctor both believe that to be the case.
Well, it is a little hard to guess because
To put it another (less spoilerific) way...
However, there seems to always be two ways to defeat Superman:
The problem with killing Superman isn't that his body operates generally so utterly different from that of a human. If you could shoot a bullet through his brain, he would probably experience brain death and die. If you lacerate his skin enough to cause blood loss to the point of no return, he would probably die. And by probably, I mean almost certainly until some out-of-universe writer put in some in-universe explanation as to why he did not die.
However in order to get that done, you would have to get around his near-invulnerability and incredible regenerative properties.
So my real answer is:
Yes, but good luck getting there
My gut reaction would be to say, yes, Superman would die from massive blood loss. But I'm not sure that's right.
People die of blood loss for two reasons: 1) lack of oxygen to the brain. Breathing (inhalation of oxygen + other gasses) causes red blood cells in the lung to pick up oxygen which gets distributed to the body. With severe blood loss, people experience anoxia -> brain death/death. 2) Cardiovascular collapse. With decreased bloodflow, oxygen carrying capacity is reduced to all vital organs, blood pressure falls, not enough oxygenated blood feeds the heart, and a fatal arrhythmia sets in -> no blood flow to the brain -> decreased oxygen -> brain death.
Having said that, Superman is different from us. First, he does not seem to need oxygen to survive. If he can fly through space for extended periods (as I seem to recall him doing many years ago) and not suffer from oxygen deprivation, he doesn't need blood for the same reasons we do; it must serve some different function in him. Certainly the gasses in his blood (if there are an) don't obey the laws of physics: there is no expansion of gasses in his bloodstream in space, the ultimate "bends".
So, for these reasons, I can't say what would happen to Superman from massive blood loss since oxygen carrying does not be an essential function of his blood, nor does nutrition or elimination of wastes seem to be vital for him.