Ok- Housebreaking and the crate issue:
Try a reverse Pavlov on her. But a few bells on a string and hang them down low enough that she can get to them. Everytime you take her outside say, "Lilo, ring your bell to go outside/potty/whatever you tell her." Take her to the bells and put her paws up and ring them with her. We always did it in a high, fun voice so Casey knew that this was a good thing. For us, it took three days before Casey hesitantly rang her bell and sat at the door. She went out, did what she needed, and came back in to much love and adoration.
Initially, you'll need to learn when they're being coy and just rinignt he bell for attention and when they really need to go out. But, until you learn, EVERYTIME she rings that bell, she goes out. It's her way of telling you she has a need. I swear by it, as does everyone else we've told. We had Casey trained in 10 days.
I tell you this becuase I suspect the reason she is soiling her crate (usually the first thing they stop soiling) is becuase she does have a need and can't tell you. Instead of doing it in the house, where's she's gotten in trouble for it before, she's going to her crate- HER house. Need to break that ASAP.
The running-
She's being a puppy and testing boundaries! She's a girl, what do you expect?!
Does she like her leash? If not (Casey hated it), just put it on her and stick it through your beltloop. This is, she's learning to stay with you, you can keep an eye on her, and she's adjusting the leash as a part of her everyday experiences and not just when she's going out or going to the vet (ie, high excitement times). You probably want to start working on sit with her so that she does know whose in charge and what her response should be when you do need to put the leash on her.
Right off the bat, we taught Case that the command "Leash" means to lay down on all fours. For us, it was important that a child (lightyears away, but she will be around when we have babies) could easily put a leash on her. This has proved incredibly beneficial at dog parks as we watch others struggle to get the dog to sit still long enough to clip them up and take them out. It's also great for vet visits....
Edit: Your dog is a lot older than I thought...
I crate trained my never before crate trained 8 year old dogs for my Mom. I spent a week with her and noticed that the three of them (one of them is my step-dad's dog) would line up and pee one of top of the other becuase of the change in household dynamics. I couldn't deal with it-and I know she was at her wit's end.
If they constantly associate the crate with time out or alone time, it's not good. It's the opposite of what the crate is for. Try pulling into the middle of the family room and putting Lilo in there while the family is around and watching TV or reading or cooking, etc. Definitely invest in a Kong and a jar of peanut butter of cookies that fit in them. Leave it uncovered and just let her be in her own little world with her toy and with the family around her. You've got to get her to the point that she wants to be in her crate and that she knows its her home. Most properly crate-trained dogs love their crate and recognize it as their place. You can also draw the distinction between sleep time and "self time" in the crate by covering it so that she can't see out when you want her to rest and leaving it uncovered when you want her to self entertain.
