How does "Dodge" really work?

For starters, what do the dodge values represent?

The dodge value is the chance in percent to take no damage at all when hit.

Secondly, when can you dodge?

You can always dodge.

Lastly, what can you dodge?

Melee attacks are dodged just like bullets, however the player is always pushed back despite not taking damage.

There is a very very long guide available in the steam workshop, everything you need can be found there. In your case dodge can be found under Armor stats. (The loading could take a bit because of the length of the guide so dont worry if it wont be loaded as fast as expected)

While I normally don't play dodge builds, I'll answer as best I can:

what do the dodge values represent?

They are percentage chances to avoid all damage of any attack.

when can you dodge?

Any time, however the dodge chance is dependent on your stance. See answer for some examples.

what can you dodge?

You can dodge all ranged projectiles and the damage from melee attacks. I'm not sure about explosives or cloaker attacks, but I suspect cloaker ninja-kicks aren't actually attacks, but a special action, and thus cannot be dodged.

Source and more info here.