Thank you for taking in Lizzie. It is a bit of a shock, the difference between the breeds. Additionally it appears that Lizzie has a few problems to boot.
I wish I had some answers for you, but I've been blessed with dogs with few problems (other than potty training
)
The agility training sounds like a good thing for her; it will help burn off some of her excess energy. Also, I think some of her possessiveness may wear off when she figures out she's not going to be handed off again.
Some of the folks here are a real resource for YT problems. If you don't find the answers here in the "General Discussion" forum, try some of the others, like the one for training. I think there's even one for agility. Lots of different forums for different types of discussions.
Meanwhile, I hope things even out for you. I've never had an IG - many other breeds, but not any Greyhounds - however, I simply *adore* Yorkies. They're truly marvelous little dogs.