.How do you calculate penetration without an addon?

Find all the sources of pen you have and add them up. Even add one are not perfect though, right now all that I have seen give a flat value to maces, that is based on your physical resistance, and that is wrong.

.What is the penetration cap?

There is no limit to how much pen you can stack, it does not pay to stack past your targets armor level though, in pve, overland mobs have 9100 and dungeon/trial/arena mobs have 18200. Past that you will now raise your damage.

. Do healers use penetration?

Sorta. Healers focus on other things first then pen. Healer do do damage but that is not their focus.

. Do debuffs like Major and Minor Fracture factor into how much penetration one has?

They are calculated first. Then percentage based pen is calculated off that and finally, self pen is subtracted. Like this, for pve-

Physical Mitigation =(((Target Physical Resist + Flat Physical Debuffs)*(1 - %-Penetration) - Physical Penetration)*(-1/(50000)) + 1)

. For example, does it benefit a Dk to apply major fracture from Noxious Breath before using skills that stack penetration?

Noxious breath applies major fracture, so no, it will not benefit from applying another source of major fracture before.

. does it help the group or does the extra pen only help the Templar?

Power of the lights minor fracture and minor breach help the group.

. If one is at penetration cap, does extra penetration provide any benefit in pve or pvp?

Again, there is no cap to the amount of pen you can stack, outside of the amount that is in the game. In pve it does not pay to have more then 18200 and in PvP, players can have way more than that, 33000 resistance is the cap that damage is reduced but you can stack resistance past that and the player will have to penetrate past the amount of resistance you have to do more damage.