Okay let me get this straight. H is your friend and lover. G is your your partner in cohabitation and parenting as well as your lover.
G has betrayed you, lied to you, treated your feelings about these events with annoyance but you're only upset with H because she wants all three of you to talk about some issues without a one on one talk with you first?
Hopefully those issues are about why such a great friend and lover would help your other lover pull all this crap on you. H can't make G not behave this way but H could have told G to find someone else to do them with if he just has to do them. The fact that H won't talk to you one on one before a meeting with all three of you seems to be the much smaller issue at hand.
And that H has a roomate that is upset about her hooking up with all her roomate's friends makes the fact that G isn't using protection with H a HUGE red flag.