You need to move back into your own home immediately -- into your own bedroom, and into your own bed. If she is uncomfortable with that, then SHE can move out (altho tell her she is perfectly welcome to stay). Of all of the controversial topics around here (exposure, "snooping," etc.), this is one that always gets near-100% consensus: DON'T MOVE OUT OF YOUR OWN HOME.
It also has potential legal consequences, as she could get you for "abandonment." If you haven't already, you should contact a food family law attorney, preferably one who specializes in "men's rights" and paternal custody issues, and get at least an initial consultation to better understand your rights and responsibilities in your state.
NO, you should not be doing her little "honey-do" lists when you come over. She has chosen to run away from your marriage, and you are not there to be her personal handyman. If there's something that affects the kids' safety or something, or if THEY ask you to do something, then sure, but otherwise, no. Women tie their feelings of "love" VERY closely with their feelings of RESPECT, and right now, she doesn't respect you and doing things like that for her are only going to make it worse.
This is probably exacerbated by you being out of work right now.
If she asks you why you didn't do the things on her list, tell her you didn't want to, and you were busy doing thus-and-such (watching a movie, etc. -- preferably something with the kids). If she pushes you, you may want to say "I'm not the one who's trying to end our marriage here, and frankly I've decided that I no longer want to do "husband" things for you if you don't want me to be your husband. I have too much respect for myself to do that."
Have you checked to see if there is OM? Sure has some of the warning signs.
btw, I told my wife -- after several references by her to the same "I want to be best friends" thing -- that if our marriage ended this way (by her cutting and running, having an affair, and refusing marriage counseling), that I had no intention of being her friend, much less her best friend. This had a very dramatic effect on her, and she told me later when we reconciled that missing my FRIENDSHIP was a huge part of her decision to come back to the marriage.
