Arduino Hardware Issue
There are two issues here:


The "L" LED lighting. On the R3 Uno this is driven by an op-amp and if pin 13 is "floating" the op-amp may, or may not, turn the LED on. So this is normal.
The 8-pin chip near the power plug getting very hot. This is definitely not normal. You must be consuming a lot of current to do that. You could try measuring it. A Uno running an "empty" sketch should be consuming around 50 mA. Note that this chip is the op-amp mentioned in my first point. Possibly you have damaged it. Did you connect an LED (or motor) or similar to pin D13? If so, what current-limiting precautions did you take?





Try running an empty sketch (or the Blink sketch) with no hardware connected to the board, to see if it still happens.
Try measuring the power consumption, if possible, with a meter measuring the input through the 5V pin, if you have a 5V power supply you could use for this.
Are these genuine boards, or cheap clones from eBay?