Yes, Elemental Affinity will only affect fire spells (given your choice of gold ancestry)
The Draconic Bloodline feature, Elemental Affinity, says:
Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, you can add your Charisma modifier to one damage roll of that spell.
Where it says “associated with your draconic ancestry”, it is referring to what you get from the sorcerer archetype (Draconic Ancestor, which has the Draconic Ancestry table), not your race.
This is confusing, however, given that the Dragonborn racial trait is also called Draconic Ancestry.
Yes, you have resistance from your race and class (although the latter comes at a cost)
Dragonborn have damage resistance, which is linked to your Draconic Ancestry from your race, which in your case would be cold damage.
You can also gain resistance from your class (fire in your case), from that Elemental Affinity class feature again, but it only lasts an hour, unlike your racial resistance; quoting from Elemental Affinity again:
At the same time, you can spend 1 sorcery point to gain resistance to that damage type for 1 hour.