Can I change the Web Player's settings or is it a one-click program? I have problems with default screen resolutions.
Yes, there are settings available to be configured, but you will have to do it through the web. The developers offer a page where you can configure the settings for the Web Player. This way you change the settings for the plug-in installed in that browser. Use the following URL to access the settings menu.
Settings: https://unity3d.com/webplayer/setup