Re your H's mental well-being... it does seem he is in a dark place and it has got to be really difficult to parse out all the things you're feeling, the anger/hurt/betrayal for what he's done, the concern/worry for someone you care about (or even just the father of your children, hoping it doesn't affect his parenting) and all the conflict inherent there. Ugh. All I can say is you seem like you're really doing well, able to recognize all of this while it is happening, protecting yourself so you can be the best mom you can, while still having empathy for him as a human being. It is pretty amazing-- YOU are pretty amazing.
From all I've read, being sure your kids know they can talk to you about ANYTHING is really important now-- you don't want them to not tell you something because they don't want to upset you, make you mad at dad, etc. Practice your validating skills on them! Also, if you haven't read Wooba's thread, especially earlier on, you might-- her H struggles a lot with depression and alcoholism and she had some times where she was legitimately worried about him, suicidal comments, etc.
On the gifts.... ooooh, girl. That burns me just thinking about it. Seriously. Also... WHY would she do this? That is just nuts. I agree with the others-- it is not within your control, but you are under no obligation to get them to write thank you cards. I don't think your H is in a place where any conversation about the inappropriateness of that behavior would be productive. I would do my best to ignore it. And if the gifts eventually get lost or broken, oh well.
As an aside, if I were you, I would stop calling it an EA or her an EAP. Maybe this is just me, but I know at least for me when I thought it was "just" an EA and then found out it was a PA, it changed my perspective somewhat on everything. I felt like when I thought it was "just" an EA it was somehow less of a transgression and the "EA" term minimized it somehow. I think here on the boards there is a bit of that too, the idea that an EA is not as bad or more forgivable than a PA. (I certainly know my H thought so, which was why he lied about it for so long.) Whether it was "just" an EA or a PA... I don't know that it changes anything in terms of his behavior and his choices and what they mean for you, unless the PA is a bright line for you. Would it make a difference for you?
Anyway. I think I said this earlier, but when you had your mini-reconciliation, it sounded like his guilt was so deep that (not knowing him of course, but) it seemed excessive if he hadn't crossed the physical line. And, the fact that she is sending gifts to the kids... that is so weird and seems like they might have some happily ever after fantasy going on here. I'm not trying to be a pessimist, just want you to be prepared.
And... gentle 2x4.. the AP didn't destroy the kids lives as they know it. Your H did. She was a partner in it but he is the one who has made and is making these choices. She means nothing, truly. (I am saying this but also to myself as I struggle a lot with demonizing the AP rather than facing the fact that my H is truly more responsible than she is.)
Of course you don't want to tell your children that their father is ruining their lives, so much easier to blame her... but she's a wacko who thinks it is OK to have an affair with a married man, destroy his family and potentially her own, and send his children presents. (Because, you know, true love.) She is not worthy of even your contempt. If he was thinking straight, he wouldn't let her do this and put the kids in this position, but he's in la-la land. You'll just have to be the strong one, the bigger person, and know that you're doing the right things for the right reasons.
DejaVu... holy cow. I have so much admiration for your strength and ability to rise above in order to do the best you possibly can by your kids. Thanks for sharing-- it is really good for all of us here to see what life can be like on the other side.
Thinking of you, Sage. xx
Sending a virtual hug.
Me (46) H (42) M:14 T:18, D9 & D11 4/19 - 12/19: series of escalating BDs 9/20 - present: R and piecing
Also, if you haven't read Wooba's thread, especially earlier on, you might-- her H struggles a lot with depression and alcoholism and she had some times where she was legitimately worried about him, suicidal comments, etc. 
