I'm reading the wounds section and I'm not grasping the idea of how they work.

Does a player get a wound card for each damage he has suffered or for each attack that caused damage?

In other words:

John receives 10 damage from an attack.

5 of them go through (when you calculate armour + toughness).

Are those 5 called wounds or are they still called damage?

If they are called wounds… does John receive 5 face-down wound cards (assuming non of the damage was critical)?

The 5 that get through are called wounds and John would draw 5 facedown wound cards and place them in front of him.

Above is correct. It is very important to distinguish damage and wounds.

Damage is reduced by soak to arrive at wounds.

Some effects/cards specifically say "inflict/suffer etc. wounds", meaning skip soak and deal those wounds directly.

Also wounds "count up" and have no limit. If your wound threshold is 12, you can have 20 wounds. You are unconscious and you are 8 away from being conscious (maybe more than 1 night's rest).

You should post these questions in the Rules questions section of the forum.