I don't think they're actually Karen's kids, it says their bio mom left their dad, and Karen was likely flirting with their father and he thought she would marry him and become the kids step-mom.
I'm pretty sure Frank is the dad but Karen isn't the mom.
Since Karen not-being-their-mom was already mentioned, I wanna focus on your other point. I also agree. I don’t think adopting them is the best option. I guess, if this was real life, I would be all for it since Elsa and Rolf since they can be good parents but in this story, I want it to focus on just the two of them with Wilbert and Hugh, and whatever drama the author plans to throw in. If the the two sisters were to be adopted, they’ll fight for screen time and I’m actually not sure how the plot will progress with them being introduced this late in the story.
Call he heartless but I don't want Elsa to adopt these children if that's where this story is going. Her sister's children is not her responsibility. What if Karen has more children our of wedlock out here, will she take in ALL of them? I'd rather Elsa build an orphanage and make the siblings center of it.