Is it possible when I create a message compose view controller to a specific number that I am already engaging a conversation in iMessage or text, that I continue the conversation?
No. That compose VC will not show conversation and there is no way to do it manually, but once you send message to recipient XY and then open Messages app, you can see the new message in conversation with XY. So iOS will automatically include this new message in existing conversation, but not in this compose VC.