When rolling extra dice for a range 1 or range 3 attack, does it go by the ship's range inside the firing arc or the closest point of both ships regardless of the firing arc?

Meaning, Ship A is attacking Ship B. The portion of the base of Ship B that is inside the firing arc of Ship A is at range 2. However, because of the way the ships are angled, if you measure from a spot outside the firing arc of Ship A to Ship B the range is only a 1. Does ship A get extra attack dice?

So does range only matter inside firing arc or is firing arc only for seeing if an attack is possible and range is ANY closest point between the two ships?

If you're using a ship with just the primary arc (x-wing), you measure the range only inside the arc, outside doesn't matter.
The only case is with turrets (or Hound's with it's auxiliary 180º), you ignore the arc and measure the closest point. You only check the arc to activate abilities (like autothrusthers).

The only case is with turrets (or Hound's with it's auxiliary 180º), you ignore the arc and measure the closest point.

The Hound's Tooth is not an exception. You still have to measure to the closest target point that's within the arcs your weapon can fire into.

When rolling extra dice for a range 1 or range 3 attack, does it go by the ship's range inside the firing arc or the closest point of both ships regardless of the firing arc?

In general, it goes by the ship's range inside the firing arc.

Meaning, Ship A is attacking Ship B. The portion of the base of Ship B that is inside the firing arc of Ship A is at range 2. However, because of the way the ships are angled, if you measure from a spot outside the firing arc of Ship A to Ship B the range is only a 1. Does ship A get extra attack dice?

No, assuming ship A is not attacking with a turret, it will not get an extra die.

So does range only matter inside firing arc or is firing arc only for seeing if an attack is possible and range is ANY closest point between the two ships?

One nice way to see it is that you determine a "line of attack".

For a standard attack (say, from an X-Wing), this "line of attack" will be from closest point to closest point within the attacker's arc. If the attacker is using a turret, the "line of attack" will be from closest point to closest point, disregarding all firing arcs.

You use this same "line of attack" to check for range and obstructions (asteroids and debris mainly).

Now note that this "line of attack" concept is not an actual game term - I just made it up to make explaining easier.

Also note that certain abilities, like Autothrusters, will require arc and range checks that are independent of this "line of attack". You can read the FAQ for all the details.

Nothing new here, I just like to quote the rules when applicable. Rules reference, P4:

If attacking with a primary weapon, the target must be inside the attacker’s ring arc and at Range 1–3.

So primary weapons attacks are subject to the "Inside Arc at Range X" ruling (FAQ p4)

The only case is with turrets (or Hound's with it's auxiliary 180º), you ignore the arc and measure the closest point.

The Hound's Tooth is not an exception. You still have to measure to the closest target point that's within the arcs your weapon can fire into.

Yes, that's right, now that you said it I see I wrote bad, sorry guys