I have a texture.png which I would like to tile, to create a textured background for my window.
Will I run into any problems with different screen resolutions?
If so, what is the best way to support a textured background for multiple devices?
Should I have 1 image instead? How large should it be?
You should create 3 different versions of your texture and place them in the ldpi, mdpi and hdpi folders appropriately. You can find more on the different densities here. In order to create a background drawable, create an XML file like it is described here: http://developer.android.com/resources/articles/window-bg-speed.html
The attribute android:tileMode="repeat" will do the magic stuff for you. :-)