A few resources to get you started:
In the OM, the Best Bets are exposed as a collection off individual Keywords in the
There are some PowerShell examples of working with Best Bets here:
Add, remove and display Best Bets for a keyword by using Windows PowerShell (FAST Search Server 2010 for SharePoint)
As far as adding the Query Rules in 2013 goes, this is a nice blog post that has some example PowerShell and walks you through the process:
SharePoint 2013 - Manage Search Query Rule using Powershell
You'll have to store the values in an intermediary format to move it between servers (CSV? XML?), but I'm sure you can figure that out.
Ah, part of the disconnect here is that SharePoint 2013 did away with Best Bets.
Check out this blog by Pedro DeRose:
What happened to Best Bets? Introducing Query Rules
Also, this page explains the new changes to working with Query Rules, but from the perspective of reading them, not creating new ones:
What’s New In SharePoint 2013 Search (Developers Perspective) Part Three
You're right though about a lack of documentation on this. Digging through the OM though, there are a couple interesting classes / properties / methods that I would toy with if I were you.
I did find one questionably legitimate result on Google from a Russian news aggregator site(!?), but only on a cached version. It's a question where the poster is using the
QueryActionType enumeration. The link to the original article has been deleted from MSDN... so... I have a screenshot from the cache: