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Is there any way to forward a thread of text messages to an email address or export them to a file on my computer? I also wonder where they get backed up to?

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They are saved on the cloud (OneDrive), as you can read here

You can automatically back up your SMS and MMS messages to the cloud. That way, the next time you get a new Windows Phone, you can restore your conversations when you set up the phone. When you delete your messages from your phone, they'll be deleted from the cloud as well, so you're always in sync.

I didn't try, but it seems it is possible to export SMS to the SD card with the App Transfer my data

On supported phones, contacts and messages can also be transferred to and from an SD card.

In the App's reviews there are users that say they did it

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