You can, but it's rng if it sticks or not like with any random Nyrea or bothrioc.
Fertilizing first is guaranteed within a window of time (if you miss that window though, you'll see a random beta nyrea next time you visit who carried the eggs you sired).
Pretty sure ya can, Taivra laying eggs in you and fertilizing her are separate checks, so you don't need to knock her up first. If it's not triggering, double-check your fertility/pregnancy settings and the wiki's flag notes.