I Just Replayed Episode 3 *SPOILERS*
 

When Lilly is accusing Ben of stealing the supplies, I chose the dialog option. "I Did It." (Something like that) And later when I was in the RV, I took the pencil, and then Lilly talks to me saying that she knows why I did it, and that she wanted to kill me while my back was turned. Did Lilly actually think that I was the one taking the supplies? I then chose the option, "Please, Don't." Lee didn't even get to say anything before Lilly body slams him out the door. So does the option I Did It, even with Lilly saying she KNOWS it wasn't me, actually make her think I did it?

Also, the option to come with Lilly doesn't appear. The only options were (I think I remember correctly) "GO", "Please, Don't.", and "Why?" (I don't think the last one is right)

I think when you help kill Larry and say stuff like "It's only a few things" when Lilly talks about the missing supplies, she actually belives it.

If you keep Lilly on the RV, your last conversation with her changes due to your relationship. Did you help kill Larry in the meat locker? That might be the reason for her being aggressive.

You can have the option to go with Lilly only if you sided with her in the meat locker.

I replayed that scene awhile ago. I did not help kill her father, and actually fed him when I had food. When I took the pencil, I turn around and Lilly is there. I forget what was said next question. But I do remember she said she was leaving. I said I wanted to go with her. So she gave me the look of shock, and then asked me to go get Clem and we would go. As I walk out the RV, she takes off on us and leaves all of us behind.

If you kill Larry, she'll always throw you out in a more aggresive way and never gives you the option of going with her. If you tried to save Larry, she'll give you the choice to come with but will either drive off when you leave or throw you out if you say you'll try to stop her.