Hello. Welcome to Android Central.
I can only start this off
Individually you can hold a text until a list with forward appears, then select a contact's email or type the address in. I only discovered I could text to email from a thread here last month.
Full conversation view I don't know.
Alternatively, I use SMS Backup+ which puts all my texts by date in a folder on my Gmail inbox page. Very useful.
I am told that another app is even better, SMS Backup and Restore. It does a conversation view and I understandably was confusing the two, but backup + works great for me. (Note 3).