What does giving lutefisk to the lutefisk god actually do?

If you tithe enough pieces, you get an artifact you can use or sell.

From the wiki:

Each Lutefisk tithed increases the chance of receiving a random reward (each Lutefisk increases the chance of obtaining an artifact by 0.2%, up to 100%).

You can sell to the shopkeeper three ways:

1. Click on the item in your inventory and then drop it onto him.
2. Click on the item in your inventory and then drop it onto an available space in the shop.
3. Hold down shift while in a shop to enable quick-sell mode. Then click on the item in your inventory. It will be sold immediately without telling you how much you will get or confirming the sale, so be careful.