Dear dlh, I am so sorry for your obvious pain and heartbreak, as well as the understandable confusion. I am also sorry for your children. This is so disillusioning.
You can learn a lot from this site and Dr. H's books. As for the immediate questions you asked, the best thing you can do is be kind and be yourself as best you can. You want your H to see you at your best, to realize who you really are, though he may not see it through this fog of his infidelity. Most affairs die a natural death within a few months. Your best advice, in terms of Harley's general counsel, is to wait it out as best you can. This is very, very hard to do.
Of course you won't act like your H coming home to collect his belongings is a natural and normal thing to do. If asked, you would honestly respond that you would like to work things out with your husband, and that that cannot happen when he moves out. But if he doesn't ask, let him go.
I really recommend that you read Surviving an Affair and His Needs, Her Needs. I am not qualified to give advice, we are all in the same boat here at the site and we're just trying to salvage and improve our own marriages.
I do feel for you and wish you well. You are going to need to spend some time on yourself, ask yourself about your emotional needs, your husband's and whether or not you've been meeting those needs. You mentioned nursing your husband through his recovery from surgeries. How about his other needs? You are going to need to be strong. Do you have family support? A friend to lean on? Come often to the site for understanding and support.
I hope you will find your way through this difficult time. Take care.