Yes, that's all approved.

And indeed, there's no way to increase Birthrights with points. They normally increase over the course of play in various ways: non-supernatural followers accrue as your profile in the World increases, some relics are captured (nearly every titanspawn and titan has a trophy relic of some sort), and gods give you more powerful creatures, guides, followers, and relics as you become more powerful, and thus more useful to them.

For one example of a way to increase Birthrights, Elena indicated she could readily give Mateusz a relic that will let him channel the Star Purview.