Yet Another Weather Plugin (Yawp) allows you to retrive weather information from openweathermap to your website
You can do this by editing yawp.css in plugins->yet-another-weather-plugin->css->yawp.css. Customizing background through settings page is coming in the next update.
Yes you can. Unpack your icon set to plugins->yet-another-weather-plugin->images and rename your icons to match the name of the current icon for corresponding weather type. Note: your icons must have .png extension.
Please email to email@example.com and I will be glad to help you out. Note: please include in the email your WordPress version, Yawp version and whether or not it works with WordPress standard theme Twenty Twelve. It will solve your issue much quicker.