Monster cards do not have life points and there is no concept of 'damaging' a monster card except for battle. In battle a monster either dies or it does not, there is no accumulating damage such as in Magic the Gathering. A monster can fight off an infinite amount of monsters that are weaker than it, but is destroyed if it battles with one of equal or greater power.
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Your monster is destroyed if the monster you attack is in attack position, and you lose life points.
When a monster attacks a defense position monster with DEF equal to to the attacking monster's ATK, no damage is inflicted to either player's Life Points and no monsters are destroyed in battle. Some monsters have effects that activate before or after they battle, so you must take those into account as well.
There are many ways to lose life points. Every player starts out with 2000 at the beginning of the duel. Some cards say to inflict this much damage to your or your opponent's life points often after a certain requirement has been fulfilled or to pay as a cost for a certain effect. The most common example of losing life points though, would be if one player attacked another player with a monster with atk points higher than the opponent's atk position monster. The difference between their atk points would be inflicted to the person with the lower atk point monster.
Yes. If your monster is in attack position and you attack a monster that has more attack points than yours, your monster is destroyed and you take Battle Damage to your Life Points equal to the difference between your opponent's monster's Attack Points and your monster's Attack Points.
No not unless as a result of a monster effect, spell card or trap card.