I created the app HangoutNow to do this. Due to the way Hangouts is designed, you still have to press the Call button in Hangouts Dialer to start the call. I am working on the next version of the app which will most likely be able to press the button automatically.
Same for me. Looks like they're taking another step to remove it. Meanwhile chat.google.com looks like it will be a web version of Hangouts Chat so maybe when they open that up for consumer use, it will function in a very similar way to hangouts.google.com's webapp interface.
If you have Android Messages and Hangouts on the same device it should be able to see your SMS messages already. This will only work for SMS messages, not Hangouts conversations.
There are SMS backup apps like this one that allow you backup your messages into XML files. Titanium Backup can do this, too.
I use Fliktu to manage my "Share" selections. The dev was active on /r/android and gave out free codes, but it's well worth it in my opinion.