I have an image that I found on another site. It's just a one pixle wide stripe, and I'm wondering if it's protected by copyright.
That image can't be copyrighted. In order for something to be copyrighted, it has to be an "original work", or be something like a book or an image that required creativity, imagination, etc. to create. A stripe can be created by anyone, without even thinking about it, so it does not fall under copyright restrictions.
Semi related: if you are going to try to use an actual picture as a background, one that is protected by copyright (see above), you should make sure that your allowed to do so.