First-hugs. You're hurting and feelings are what they are, when we hurt, we need comfort.
Second-There's no right answer for whether or not what you are asking is fair. It completely depends upon your needs and his and the terms you agree to.
BUT-I understand your feelings somewhat. I went through similar this winter with my husband. Not the same, just some similarities. Particularly the nausea and depression over the same examples you gave.
I highly suggest scheduling a day a week that there is nothing-no reminders in front of you-about her-for the two of you to connect in your own relationship. That time to be used NOT TO DISCUSS THE ISSUES-but to connect in love and rebuild that solid foundation for yourselves.
Then-you need to really address what your "hard limits" are regarding the situation. BE SURE TO CONSIDER the limits you think you want-with the tables being turned; as in "if i was in love/connected to someone deeply and he didn't like them-would I be ok with him telling me xyz limit was required for OUR relationship to continue".
Then-time to sit down and lay out hard limits-HIS AND YOURS and figure out if there is a way to compromise so you both feel satisfied.
I will say-that once my husband understood my need for him to lay out for the other lady the misperceptions he promoted in her regarding me AND he was willing to make that commitment to having one day a week where I heard/encountered no reminders of her, I was able to work through my own feelings and at this point have no issue with them having a friendship (they opted out of dating) because I know he and I are back on track with one another and that's the most important part.
I would be ok with them dating-that they aren't is their choice-not mine.
"Love As Thou Wilt"