I think that its perfectly normal. My daughter did, and still does most of the stuff that you described, and she is 3. Some days, I can leave the house without a hitch, and others, ugh. Now, I ask her to take care of her baby sister, that idea has seemed to work so far. For the seperation anxiety, have you tried to play peek-a-boo with her? When you leave the room, just peek your head out and say do the boo. My daughter would just start laughing, and each time, wait longer in between the boos, and you can always reassure her that mommy is coming back, and keep on talking to her, at least if she cant see you, she can hear your voice.
For the eating, its the age. It's basically go with the flo with that one. They get so into playing and what they are doing, they forget to eat. If you still have concerns about her nutrition, there are children's vitamins, or talk to your pedi when she goes for the 18 month visit.
As for the sleeping, like the previous poster said, she isnt gonna being doing it when she is a teenager. Enjoy it while it lasts. Instead of using the gates,(my little girl can climb over it easily) I got those plastic child-proof doorknobs, and i have it on the inside of the room, so she cant get out. I'm actually looking forward for when the baby gets old enough for the 3 of us to bunker down together.
Unless your friend has a Phd in any type of psychology, I'd tell her thank you for your concern, but I am doing a fantastic job being a single mom.