If the clock is an active icon (and not like "iOS6 calendar active"), I can't imagine the weather icon WOULDN'T be.
Clock does not need to use location (GPS) to give you live information. But weather app needs to. Resulting on a battery consummation. At least this is what i think.
Not if they give you the option to set it to one of your favorite locations, not just the current one. They can also give you the option of how frequently you want to update the info.