All that the rules say is the base must be unoccupied for the runner to advance with a dropped third strike. Now, in your situation there are two outs which changes things a little. It allows the runner on first to vacate the base by attempting to advance to second. If there is a runner on first with less than two outs the batter cannot attempt to take first on a dropped third strike no matter what the runner on first does. In your scenario as well, the bases are loaded so all runners must leave their base in order for the batter to advance to first making it a force play at home plate.
All that the defense needs to do is step on home plate with the ball before the runner from third can successfully cross home plate in order to record the strike out.
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Only on a third strike with 1st base un-occupied. This does not apply to Little League.
Note: he is entitled to TRY to run to first base. He can be thrown out by the catcher if the throw reaches the baseman before the runner. If he beats the throw, then he is safe.
The pitcher is given credit for a strikeout. The catcher is given an error.