1.
The frog population doubles every day, so the pond was half full of frogs one day before it was full. That means two days before it was full of frogs it was one quarter filled with frogs. Thus, on the 14th day the pond was one quarter full of frogs.
2.
Turn on switches #1 and #2. Leave them on for several minutes. After the minutes have passed, turn off switch #2 and leave switch #1 on. Now open the door and go in the room. The light that is still on is obviously switch #1. Whichever of the other two light bulbs is warm to the touch is connected to switch #2. The light bulb that is cold is switch #3, which was never turned on.