I'd be more than happy to assist you.
1. Your screen is most likely going dim when you're reading a web page but not actively clicking around. It's sort of a time-out feature. You can adjust this under SETTINGS, DISPLAY there should be an auto brightness setting, or a display timeout setting.
2. The task manager stops programs running in memory. It won't delete software fro your phone, it just kills it so to speak. If your phone gets sluggish you can go in and kill old apps that you were using earlier but are no longer using. Android phones have pretty good memory management built-in, but sometimes you need to manually kill an app. That's all it does, it just kills the app, doesn't uninstall or remove from your phone.
3. The large 1 means your app is trying to get your location for weather but there was an error. You can allow this app to pull your current weather based on your location, or just enter the app and set your zip code. The 1 is most likely an indication that your exact location wouldn't be honed in on and it's using another method to determine your location (i.e. cell phone tower triangulation etc) and thus has an idea of your location but not the exact pin pointed location.
The Galaxy S III is a brand new phone but the questions you've asked also applied to the Galaxy S II version. So while I'm telling you where to go to adjust settings that's based on the Galaxy S II version so you may have to sort of read between the lines as the exact verbage/labels may be slightly different.
To adjust the Weather app, you should be able to click the weather icon. Once it launches, click the MENU button to access the settings of that app, change the GPS or option to get weather based on your location and try entering your zip code instead.