Where to start? Many women (and men) get the Mommy Syndrome. You know mommies aren't supposed to be sexy, they are supposed to be, well, motherly? That can cause some real issues.
It does sound like therapy is a good idea, she's had some difficult stuff to deal with that could have been very damaging to her self esteem.
The working late going out to dinner with another man does sound iffy. Can't blame you for being concerned. At least she has told you, that's one positive in there.
Kids can really be stressful and yes, they do reflect your stress and often magnify it.
All this inviting people and such is an avoidence technique. She is avoiding intimancy. When you have sex, does she orgasm? Are you sure? Interestingly in surveys and studies men pretty universally say they can tell if a woman is faking it and that their partner(s) have cum. Yet when the women are questioned almost all say they have faked it at some time and some fake it all the time. A woman who isn't getting full pleasure from sex, isn't very motivated to keep having it.
Over and over we get posts her about the pill causing women to lose their sex drive, so that may not help much. A non hormonal IUD would be a better choice in that regard. It's is interesting that she made this change without any discussion. Did she give any reason?
I would guess that your performance is suffering because you are stressed. It might not hurt for you to have a few visits with a therapist. They may be able to give you some insight and some ideas. It certainly wouldn't hurt to give it a try.
Being a pushover isn't the answer and neither is goinf overboard on the romance, you are right, that could push her further away. Communication is really your biggest tool right now. Let her know how much she means to you and how much your marriage means and that you are ready and willing to do what it takes to save it. See if the two of you can find some sort of middle ground?
I have to admit to getting very frustrated with these situations. I've been there, trying to hold it together - more than once (I'm a slow learner I guess) it is very painful and difficult. The only person who's actions you can control is you. You can't make her respond as you want. You can't make someone love you or want you, no matter how badly you want to. Focus on yourself and your daughter and let your wife know you are open, loving and want to find a way to make it work.