You are quite right, Agostin. These verbs and related nouns and adjectives are quite close in meaning and use, but there are a number of distinguishing characteristics.

hurt (verb)

If part of your body hurts, you feel pain there. If you hurt someone, you cause them to feel pain. Note that verbs that refer to physical feelings (hurt, ache, etc) can often be used in simple or progressive tenses with no difference in meaning:

• Have you been knocked over? Tell me where it hurts / it's hurting. ~ My arm hurts.
• You're hurting my arm. Ouch! Don't touch me. That hurts!

You can also hurt someone's feelings, and cause them to feel emotional pain:

• I think she's going to be hurt. I don't think she'll ever fall in love again.
• What hurt me most was the betrayal. How could he behave like that?
  

hurt (noun/adjective)

• The hurt that she felt was deep and would only be softened with the passing of time.
• They were suffering from shock but did not seem to be otherwise hurt.
  

injure (verb)

In the sentence describing people suffering from shock above, hurt could be replaced by injured. If you injure somebody, you cause physical damage to part of their body usually the result of an accident or through fighting:

• A number of bombs have exploded, seriously injuring scores of people.
• The demonstrators injured a number of innocent people when they started throwing stones.

injured / injury (nouns) / injured (adj)

• The injured were taken to hospital by air-ambulance.
• Their injuries were thought to be serious.
• He was not seriously injured, though his coach took him off at half-time as a precaution.
• Two minutes of injury time were played at the end of the fist half.
  

wound (verb)

If you wound somebody, you inflict physical damage on part of their body, especially a cut or a hole in their flesh caused by a gun, a knife or some other weapon, often in battle.

• There was no escape. They were mortally wounded by the enemy fire.
• The driver of the Red-Cross ambulance was wounded by the shrapnel.

In English, it is often a matter of knowing which adjectives collocate with which nouns and which adverbs go with which verbs. In this particular word family, the adverb-verb collocations are normally as follows: badly hurt / seriously injured / mortally wounded. You will also have noticed that with these verbs the passive voice is often used.

wound (noun) / wounded (adj)

• The open wound really needed stitches and took a long time to heal.
• The four wounded men were taken to the field hospital in the back of the Jeep.

We also have the expressions: to rub salt into the wound, i.e. to make an unpleasant situation even worse and to lick one's wounds, i.e. to slowly recover after being defeated or made to feel ashamed or unhappy:

• I didn't want to rub salt into the wound so decided not to mention Bob's infidelity.
• The British team could only retire and lick their wounds after such a comprehensive defeat on Spanish soil.
  

damage (verb)

It is things that are damaged, not people. Damage is the physical harm that is caused to an object. More abstract qualities, such as reputations and the economy can also be damaged. Compare the following:

• The car was so badly damaged in the accident that it was barely worth repairing.
• When he got home, he discovered that the vase he had bought had been damaged.
• If he continues drinking like that, his reputation as a defence lawyer will be damaged.
• High inflation was damaging the country's economy.

damage (noun) / damaged (adj)

However, we can also speak of someone being brain-damaged (not brain-injured) or suffering brain damage. But this is an exception. Normally damage relates to inanimate objects:

• Professional boxers sometimes suffer irreversible brain damage.
• It was a huge bomb and the damage caused to the shopping precinct was quite extensive.

We also have the informal expression: What's the damage? meaning 'What is the damage to my purse or my pocket?' in other words: What do I owe you in payment for this service or these goods?:

• Thanks very much for the work you have done on those curtains. What's the damage?
  

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