If I cast Elemental Betrayal (https://2e.aonprd.com/Spells.aspx?ID=797) and choose the Fire trait, the spell can now deal fire damage. Does that cause the spell itself to gain the Fire trait?
For example, after using Elemental Betrayal (fire) could I then use the Conduct Energy ability (https://2e.aonprd.com/Actions.aspx?ID=704), which requires me to have just cast a spell with an elemental trait to it?
No. Your spell only has a choice's trait if it says it does.
You call upon the elements to undermine your foe. When you Cast this Spell, choose air, earth, fire, or water. Each time the target takes damage from a spell or effect with the chosen trait, it takes an additional 2 damage of one damage type dealt by the spell or effect. If you chose fire or water, the target also takes this additional damage when taking fire or cold damage, respectively, from effects or spells without the chosen trait.
Contrast protection:
You ward a creature against a specified alignment. Choose chaotic, evil, good, or lawful when you cast this spell. The target gains a +1 status bonus to Armor Class and saving throws against creatures and effects of the chosen alignment. This bonus increases to +3 against effects from such creatures that would directly control the target and against attacks made by summoned creatures of the chosen alignment.
This spell gains the trait that opposes the alignment you chose—if you choose chaos, this spell gains the lawful trait, and vice versa; if you choose evil, this spell gains the good trait, and vice versa.