Pac, please sit on this for at least through the weekend. You are acting on pure needy emotions. This is a time that could work for you if you are smart.
She is temp checking you, don't you get it? This is the time that you can lay out your terms of her coming back to the marital home. I am strongly against a man allowing his W to come back and them sleeping in separate beds. Now understand, most WW's (even the remorseful ones) are not quite ready for sexual intimacy, b/c it takes some adjusting to get the OM out of her head and make ready for her H. However, the two of them need to sleep in the same bed, to slowly establish some emotional closeness/intimacy. If she sleeps in a spare room, you doing a lot more than compromising. This is not being "flexible". It is giving way to what is wrong and setting an inappropriate picture of what a MR should be. This is not what your daughter needs to see and believe this is what M really looks like.
If you do not lay out your terms for her coming back, you may never get her back the way you want her. You are in for a lot of hurt by agreeing to her conditions. Be the head of your home. Set the standard and example. Be the leader in living right.
Your email sounds way too weak and opened to what she wants. You need to state your conditions. Do not say that it's her home too. Although her name may be on the deed, just don't say it. You are giving her way too much power. That's why we needed to see how you are stating the email. But all we can do is try to advise you.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!