Hi - you are getting good advice here, but I do think it matters whether it is MLC or not.
There are affairs, which are hurtful, and have their own reasons and pathology, and there is MLC in which the affair is usually regarded as a symptom of an underlying crisis. Until that is addressed by the MLCer they are not in a position to work on their marriage or anything else.
We can all talk about what they ought and ought not to be doing until we are blue in the face. My youngest son told me that his father's behaviour is just like that of a cocaine addict - not just to me but to everyone else. They are self deceptive, manipulative, self serving, and concerned totally about themselves. Bt until the addict wakes up and realises what he is doing to himeslef and everyone around him they are totally resistant to whatever anyone else will say and do about their problem
Whether or not you allow him to remain in the house is up to you, of course. But I believe that it may help you to remember that this is MLC, that he probably has a long way to go before he wakes up and starts to work on himself, and in the meantime he is pretty much resistant to reason, threats, or any attempts to restore him to himself.
Having had an H who went through a very intense MLC...who also convinced his OW to leave her H (she had 3 or 4 small children to boot)...I saw the extreme guilt he carried...and as I said it took time for him to get over her and the guilt...it also took time for him to feel comfortable with me again...and yes, there were/are performance issues and yes, I wonder if it is me...but I don't focus there long...I also said that my H made contact with OW about 9 months after he was home...he also profess his "love always" to her...thanked her for "saving my life"...for telling me that "you always make the right decisions"...but reading between the lines I know what he was doing now...he was easing his conscience with her...he didn't want to "reject" her...but he fished enough to find out that she had moved on and found someone else and was now living with that man...she was happy...to my knowledge that was the last contact he had with her...he also expressed his love to me a few months following that contact...and from that point things have been improving on a steady basis...it has been 2 years now since his return...it has taken a lot of time...
I too believe my H returned too soon...but looking back it allowed me to show him the real changes I had made in myself...the strength that I had gained...I had to deal with his lack of love for me...lack of intimacy...alcohol addiction...severe depression...and inability to hold a steady job...this wasn't a pretty picture....and believe me I had many doubts...but in my heart/gut I always felt that if I was strong enough it would work out...if it didn't...I would be OK...
I understand how you feel...and your right, until OW moves on your H probably "CAN'T" commit to you...it is tough place to be...hopefully soon either she will be gone or he will give up on her...but really all you can do is be strong...avoid conflict with him...and move forward for yourself...and pray!