You have a strange situation. You've "thrown her out" because she's having an affair, yet you don't have a legal separation agreement and you said you won't file. You've given her pretty much free access to the home and kids as long as she asks for permission.
Your interactions with your wife will be uneasy for some time until one of decides to end the marriage or you come to a more crystal clear temporary separation agreement.
The reason people carve out visitation/custody is to prevent the violation of boundaries and to have clear expectations on both sides. It's going to get pretty old having her call you to ask to see the kids, which leads to her "doing" things for them and then, perhaps, for you. It's going to get pretty tough for you to say no after a while.
I think she's been pretty good about asking your permission, but it's pretty weird. Perhaps it's best if you start working towards a controlled separation agreement. There's a book on the subject. It's an attempt to save a marriage, but the idea might work for your situation -- to create clear boundaries and space.
She's showing you that she is taking steps towards her independence: PO Box, paying bills, etc. And yet you lament that these are actions of someone who is "moving on." Are you, perhaps, banking on some tearful epiphany and repentance? I think if you let this "normal" behavior get to you, she'll slowly regain power and control.
By the way, she's much better at detaching than you are. Why? Because, Frank, and AmyC will probably attest to this: SHE DOESN'T REALLY CARE ABOUT YOU. Sure on some level she might, but right now, you are not in the forefront of her concern.
Does this register? She doesn't care about your feelings. She would leave you if she had the money or a sugar-daddy/rescuer. You know this.
She only cares about what your actions can do to her or affect her life. Frankly, she doesn't care about her children as much as she cares about how they validate her feelings of motherhood. She needs them to feel useful and whole. They have been her job for the last 17 years. I'm sure she loves them, but, right now, she loves them in ways that validate her feelings of self-worth.
I think her independence is a good thing. It's reality. Don't lament it. You don't really want a tearful, needy, adolescent returning home to daddy after running away. You got that last time...and what did it get you? You want a real woman. She's taking steps in that direction. She needs her independence to do one of two things: leave you or return to you as a whole and grounded person.
Ask yourself, what do you want more: the tearful needy adolescent to return home, or an independent, strong, woman to make up her own mind and make a fully informed decision? Or to be more blunt, would you rather have her home in complascent resignation and needy capitulation, only to repeat the drama again, or would you have her leave you and make it on her own?