My husband is a writer and has some professional relationships with editors and others. These are not people I know. I don't have a problem with it for the following reasons:
He does not share personal info. If someone can't get to his work because she is sick, he will ask her how she is feeling. He does not ask about pregnancies or discuss vacations (other than giving dates he will be out of contact) or send pictures of us.
He tells me about his contacts. He will share her comments with me.
He networks with a variety of people to maximize his chances of getting published. He uses LinkedIn not just Facebook.
Your husband "wants to become a writer" - what does that mean? What is he doing to advance his career, improve his skills, wide his contacts? Meantime he quit his job?
Is he telling you about all the conversations? How in the world does he justify 10-20 conversations a day? That has nothing NOTHING to do with writing! That's not how to become a writer!
Your husband is taking away from family time. If he's trying to start a second career, some of that may be necessary, but absolutely NOT if he's discussing personal things! He's kidding himself. And he's deceiving you, I think. No professional writers sign things "love" and "sweetie".
If he wants you to read every message to her so you can see how professional it is, that's good. But it doesn't sound like that. This sounds very much like an emotional affair - he says he won't meet her, so why he's doing is safe, in his mind. He's giving his emotional self to her rather than to you.
I know you don't share his love of writing, but maybe you should read more of what he does to share his interest. If he puts up even one roadblock to that, you've got a huge problem. He could be spending that time with you or with your child - or he could be spending it writing! Just giving his in-depth opinion of her writing is doing nothing to advance his career, and it's of questionable value to her in her writing career. And he's not working with any other writers or editors, correct? Big huge red flag!! I'd put a stop to this.