I want the house in Riften, but I'm not sure if the price range is just the seller trying to scam me or if it automatically upgrades part of my house. What is the price of the house and if it's a range, does paying a higher price provide any house upgrades?
You can buy it for as little as 8000 if that's all you have. It won't go higher than 12000, but the only way to get a lower price is if you only have less money. The price will be all of your money as long as it's between 8000 and 12000. Paying more or less won't affect the state of the house - no upgrades come with it either way.
(Note: The same applies to Hjerim in Windhelm, but no others.)