Is it possible to extend the inventory?

It's not possible to make the inventory bigger, just to buy more tabs for the stash. I don't foresee any changes to this in the future.

My advice to you, since I used to be the same way, is to not pick everything up, just the things that might fetch more than a few pennies back in town. If you do pick up a gray it should probably be something that you want to upgrade with an orb.