[RPG] Do I HAVE to choose a subrace? And how to add ability scores if I do

They add together. See the summary of modifiers to ability scores on page 12. And see the example of building Bruenor, Step 3 on page 13, where both his Mountain Dwarf strength modifier and his Dwarf constitution modifier are applied.

You're confusing two different game concepts, which is understandable considering someone made the mistake of giving two completely different things very similar names.

Breaking it down:

1. The Race you choose has Racial Traits. You get all of these automatically, just like you're thinking of. You pick a dwarf? You get all the dwarf racial traits. The "Gaining Traits" thing you found has no relation to these.
2. Then the Advanced Player's Guide introduced a thing called the Trait system, which you can read about there. This is what the "Gaining Traits" thing applies to. These are optional mechanical bits you pick up for your characters. These traits are sorted into various lists, like Combat Traits and Equipment Traits. To confuse matters even further, one of these lists is also called Race Traits, which are so called because they always have a particular race for a prerequisite.

The racial traits from #1 have no relation at all to the traits system and race traits from #2.

The traits from the Trait system (#2) are like flavourful mini-feats you pick at character creation and which might reflect your character's personality or background (or you can ignore that bit). It lets you pick up neat features like an iron liver or an aptness at being a tactician. The Trait system is optional and your group can ignore it entirely.