New to the forum.....About a month ago bought a 2006 H3 and so far love it....drove from near Philly to Pittsburgh over the weekend and over 245 miles per half tank. Very comfortable but have only one problem....when its cold or rainy the drivers window and power locks don't work. After it warms up no problems.
Love it

 

Welcome to HF

Take a look at the connector between the door and body for corrosion and continuity.

 

Thanks Tainter…..where exactly is that??? In the door or under the dash somewhere?

 

"245 miles per half tank"? The sender is inaccurate!
In the meantime take a look at your UFB. Make sure to be prepared:

 

There is a connector with a grey clip (retainer) between the door inner (front) and the body.
Google how to disconnect. Look for loose connections .Spray contact cleaner
Be very careful when reconnecting the connector. Don't want to bend any of the pins in there.

Also remove the window switch, clean and inspect. See if that works also.