- What is the maximum current that one GPIO output can handle
Besides the given limits per pin a total limit for all GPIO pins of 50 mA is noteworthy and should be considered. The total current sourced or sinked from all GPIO together (the sum of all pins) should not exceed the limits of the 3.3V rail. Raspberrypi.org states: The GPIO pins can draw 50mA safely, distributed across all the pins; an individual GPIO pin can only safely draw 16mA.
- Can I power LED only using GPIO output (with resistor) (with no +5V pin)?
Yes it is possible - since GPIO pins can both sink and source a current, a load can be switched to either VCC or GND. Just consider to connect the LED (with series resistor) to a GPIO pin and to GND. Set the pin to high (1) to turn on the LED.