Why do people bleed in real world when they are hurt or killed in The Matrix?

To quote from the movie script:

While Tank helps Morpheus, Neo spits blood into his hand.

NEO I thought it wasn't real.

Neo stares at the blood.

NEO If you are killed in the Matrix, you die here?
MORPHEUS The body cannot live without the mind.

In other words, even though you haven't really been shot at, or physically wounded (and thus don't have bullet holes), your mind has fully believed that those things have occurred and thus injury or death occurs in the real world.

As to the extent of these injuries, we can only speculate. As you mentioned in the comments, biting on the lips, or tongue, or other parts could certainly cause the bleeding that is seen. Intense thrashing by your body due to the trauma your brain believes you are suffering could also contribute. Additionally, your brain could actively shut down certain parts of your body (such as lungs, kidneys etc) due to the damage it believes you have sustained to them.

The brain, via stress or emotional trauma, can cause some illnesses to manifest that have no other medical cause. It's called Psychosomatic Illnesses:

psychosomatic [si?ko-so-mat´ik]
pertaining to the interrelations of mind and body; having bodily symptoms of psychic, emotional, or mental origin. psychosomatic disorder (psychosomatic illness) a disorder in which the physical symptoms are caused or exacerbated by psychological factors, such as migraine headache, lower back pain, or irritable bowel syndrome; see also somatoform disorders. It is now recognized that emotional factors play a role in the development of nearly all organic illnesses and that the physical symptoms experienced by the patient are related to many interdependent factors, including psychological and cultural. The physical manifestations of an illness, unless caused by mechanical trauma, cannot be divorced from a person's emotional life. Each person responds in a unique way to stress; emotions affect one's sensitivity to trauma and to irritating elements in the environment, susceptibility to infection, and ability to recover from the effects of illness. Physical conditions to which psychological factors are shown to be contributory are currently classified as psychological factors affecting medical condition. Any physical condition can be so classified, but the most frequently included are asthma, peptic ulcer, bowel disorders, cardiovascular disorders, arthritis, allergy, headache, and certain endocrine disorders. In recent years there has been some success in using behavior therapy to treat these and other illnesses whose symptoms are related to the autonomic nervous system. Clients are taught new ways of coping with stress and new patterns of behavior. Among the techniques used are biofeedback, relaxation training, classical conditioning, and operant conditioning using social and material reinforcement.
-Miller-Keane Encyclopedia and Dictionary of Medicine, Nursing, and Allied Health, Seventh Edition. © 2003 by Saunders, an imprint of Elsevier, Inc.

Psychosomatic reactions can be quite extensive, but for the most part, they will not cause bleeding, cuts, or other physical trauma. That said, psychosomatic seizures, can be severe enough to cause bruising, from self inflicted muscle spasms or causing the person to hit an object or wall. This is not rare in epileptic seizures, but it's also not too common.

Another way of describing it is Conversion Disorder, Somatic Symptom Disorder, or Functional Neurological Disorder.

Psychosomatic reactions have been known to cause organ failure, or sudden death. It's been coined as Voodoo Death:

Voodoo death, a term coined by Walter Cannon in 1942 also known as psychogenic death or psychosomatic death, is the phenomenon of sudden death as brought about by a strong emotional shock, such as fear. The anomaly is recognized as "psychosomatic" in that death is caused by an emotional response—often fear—to some suggested outside force. Voodoo death is particularly noted in native societies, and concentration- or prisoner of war camps, but the condition is not specific to any particular culture.

As we see in the film, Neo spasms violently in the uplink chair, in response to what his brain is perceiving. That violent seizure would explain something like biting his own lip or tongue causing blood. More of a coincidence that he was hit in the face. The sudden deaths of people killed in the Matrix is more accurate than the blood part.

Re: your edit.

Blood from the Mouth is a common trope in movies and TV. As TV Tropes puts it:

As a death trope it's caused by any fatal injury, and is less a symptom than a signal to the audience that this dude's a goner. Much like the chest-clutching and doubling over in days of old, Blood From the Mouth is a sign of death that won't horrify us with the specifics, keeping our focus on the story itself.

Specifically, Mouse, who dies a violent death in a hail of bullets in the Matrix, needed to be justified in the "Real World". Had he just died without some outward sign, it wouldn't have matched up and would seem weird to the audience. He's also unique, as everyone else has a much simpler death.

Compare that to Neo's first death. Shot multiple times in the Matrix, small seizure, then his heart just stops and he flatlines in the Real. No bleeding.

So yes, there are inconsistencies, but it's due to cinematic cues to the audience.