Is there a way to either save/copy/or email an entire iMessage conversation?
I'm only aware of how to do this natively on macOS


Sign in to Messages on OS X
Select the conversation you want to save
File -> Print -> Save 
(optional) Use whatever pdf editing software you choose to extract, copy, transform the PDF version of the conversation


There are third party solutions like PhoneView by ecamm to do this as well, but I don't know of an archive method on iOS. I'm a very happy customer of PhoneView - I sent them my money and have been satisfied by their continued updates of the product and the value.