If iMessage is off, and someone sends a message as iMessage, will she be able to receive it once she turn iMessage on? My friend doesn't have an Apple ID so she uses mine sometimes to just download apps. But me and my mom were talking about some really private stuff and we're using iMessage since she's living in another city so we don't have much time to see each other and talk about this. My friend never activated the iMessage feature on her settings, but if she does turn it on she might use my Apple ID right? 

My question is, once she turns it on, can she receive those messages me and my mom sent to each other? Assuming that the time while me and my mom were chatting, the iMessage feature on her phone is still off.
No. The messages will only pull through from the point iMessage is turned on. Only in a situation where iMessage is turned on but the computer is unable to receive them (sleep, offline, etc) will it backload messages.