Will the Arduino work safely and properly if I give a 9V battery to the 2.1 mm jack of the Arduino Mega ADK through a 150 Ohm 1Watt resistor ?
Uh, why the resistor?
It should work fine without the resistor.
Is 500 mA safe for the Arduino?
The current draw depends on the load. A 500 mA supply will deliver whatever current required UP TO 500 mA.
Can I directly connect a 9V battery to the Arduino Mega ADK to the I/O pins as an input?
NO! Just up to 5V.
Not to any pin on the chip itself. Only the 2.1mm power input jack (and make sure to get the polarity right, it's center positive, you'll damage the board if you connect it backwards)
The Uno schematic indicates a diode in series with the external power input. So at least the Uno will not be destroyed by reverse connection.
Now if I use a 9-0-9 200mA transformer (Input= 220-230V AC) which is connected to a rectifier and I connect it center +ve will the Arduino work?
If you use a 0-9v winding into a rectifier bridge and feed positive to the centre pin and you total current is less than 200mA, it should work.
But the output is AC
A rectifier turns the AC from the transformer into DC. You will need a capacitor (470uF or so) as well.
The voltage across the capacitor assuming full rectification would be 9xsqr2=12.7v
Did you take the advice in post #6 in this thread?
Hi, what is your electronics, programming, arduino, hardware experience?
If you are going into building your own power supply, I suggest that you have a VOM, and test the voltages, before connecting them to your processor. The voltage is important, the current is just what the power supply can provide if required.