Assume for a particular Pathfinder game that I am allowed to use two traits for my character. Do alternate racial traits count towards that number? I assume not since you are trading a built-in racial trait for a different racial trait, but I want to be certain.
Using alternate racial traits doesn't count toward your number of traits, and in fact they have no relation to the trait system at all.
Despite their names, races & racial traits have no relation to the trait system & race traits. I've broken down the difference in this answer; you're not the first person to be confused by it either. Safe to say they shouldn't have given the same name to two entirely separate character feature systems, especially since they both apparently debuted in the same product (the Advanced Player's Guide).
Here's how these things interact:
- Your chosen race has racial traits.
- You can pick alternate racial traits to replace any of those. This is like picking a class archetype to modify your existing class.
- Your character has character traits from the trait system, and you usually pick two for your character.
- Among the types of character traits you can pick, one of them is race traits.
The racial traites and alternate racial traits simply have no interaction with your character traits and trait limit. (Likewise your race traits have no relation to the alternate racial trait system or your racial traits.)