How did Chi end up in the trash?
In the manga, Chitose Hibiya mentions that she hoped someone would find Chii and give her all the things she was missing (e.g. Love). Chii's memory was wiped because she was sad, so this was an attempt to start over from scratch without any of her painful memories. And this also means not knowing anything about Chitose.

So Chitose put Chii in the trash so that Hideki would find her.