Is this professional communication or flirting?

Lately, my husband has been threatening me to give divorce every time there's small argument. So I decided to check his work email, if there's something that's stressfull for him. I didn't find anything stressful but following emails exchanges with a female colleague. When I have asked my husband have you heard her name, he said yes but she is not on my team. Help me decided is this is professional communication or flirting?

Email Thread 1:
Initial email from Female co-worker:
"very sorry that I couldnt connect with you today - I really wanted to enquire aboiut your family. How is your mom taking it?
It was so sudden that is unfortunate. I hope you were able to spend some time with your dad during his last days. I will catch up with you upon my return."

My husband's response:
"No worries. I know eBay life. So many things to wrapup before we go on vac.
On family front mom is doing ok now. Now I miss her more than ever. Let's catchup to refine our preprod goal once you return from vacation… Have a great vacation and don't worry about work."

Email Thread 2:
Initiated by female co-worker
"Saw your car in north campus yday - wanted to get some help from u.
will u be able to share the selling architecture and tech stack with me? I ll set up a 30 tm meeting tomorrow. Will share details in person,"

My husband's response:
"Anytime Sent from my iPhone"

Email Thread 3:
Female Co-woerker responding to a reference link email sent by my husband:
"Thanks a ton for all your help! I'm so grateful to have friends like you."
My husband's response:
"Feeling is mutual."

Email Thread 4:
Initiated by female co-worker:
": Back today. How r things? Have few mts so we can catch up?"
My husband's response:
"Welcome back. Hope you had great vacation. I'm available from 1:30 - 2:00 PM or 12:30 -1:00 PM"

I personally felt all other email chains were professional expect this one. What do you all think?

I don't think you have anything to worry about. There's no talk of meeting up outside of work or how one of them liked seeing the other one today. It's normal to have work friends and it sounds to me that's all it is.

She's showing mild interest in him outside of work but he doesn't seem to be taking the bait.

Tell him he better watch it.

She's trying it on; he's polite at best, nothing more.

The bigger issue by far is not about any communication between the two of them, but the fact that your husband threatens divorce after every small argument. I don't think the problem is outside the marriage. I think it's within.
Did you not feel a bit creepy checking your husband's email account without his knowledge or permission?

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