I want to know if Luke and Leia being Vader's kids becomes public knowledge? Presumably people would be more suspicious of them if they knew the truth.Does Poe know his General is connected to the Empire?
I don’t have any quotes, but I’ll try to give a detailed answer. In the canon novel Bloodline (Claudia Gray; 2016), which is set six years before The Force Awakens, Leia Organa is still a senator, Han Solo is still with her and Ben Solo is yet to be a lunatic.
Carise Sindian, a senator representing Arkanis, was given the title of Governor of the planet Birren by Leia, due to both of their royal heritages. Shortly after she is given this position, she discovers a message from Bail Organa that was originally intended for Leia, stating that Darth Vader is her father. Being a part of the still-forming First Order, she decides to expose this fact to the entire Senate.
Naturally, they are shocked and appalled, which results in Leia resigning her position and Carise being revoked of her title. This news quickly spread around the galaxy, so when Leia warned of the First Order in later years, most of the New Republic was not willing to believe her.
So Poe presumably knew, and it seems did not hold that against her. That’s the best answer I have. Someone can edit it if they want.