Well, the first thing we need to figure out which model and operating system you actually have. If you are running 10.6, then it's NOT a G5. So, if you can get it to start, click on the Apple (upper left corner), then on about my Mac and copy your OS information (under MacOS X) - it'll say version ..... Then click on "more info". In that window, copy what it says on the right - model name, model identifier, processor, etc.
Report back with that information.
In the meantime, the initial question is: why are you unplugging your computer? There is a small battery inside your iMac which will keep the settings when not plugged in; you should not unplug it regularly - instead, use the proper shut down method (File menu: shut down), or, you can simply let it sleep when not using it. If you always unplug it, the battery will wear out more quickly, so that might be one problem.
Second, while you're in the System Profiler as mentioned above, click on memory on the left and report what it says: is there an "ok" next to the entry for the RAM modules? How much RAM is there? Sometimes, if the RAM is defective, the Mac will make certain sounds at startup, so this information is important as well.