I presume you've got an Arduino Uno and the external power is via the 2.1mm power socket? If not please shout out as the rest of this will be based on that assumption.
I indeed googled up that one and someone suggested 10k resistor from rx to tx and I tried , didnt work.
Don't do that. I can't imagine why that was considered a good idea. On the plus since it's not a dead-short you are unlikely to have smoked anything. Rest assured putting a resistor between RX and TX is not a normal requirement - so take that out and maybe don't listen to that someone so much in future
There is nothing in the Arduino that makes it need a serial connection - unless you make that happen in your code - it's common to use those pins for purposes other than serial comms in fact, or for there to be no serial communication at all in a sketch, as it is with the blinky example you cite.
On the question of the serial monitor being open, the Arduino also doesn't wait for comms to be confirmed so if you don't open the serial monitor and there are Serial.println() etc in your code, they will just happen and be lost if there's nobody/nothing listening. You can send as much as you like without anything connected on the other side and the Arduino is fine with that...unless you code it to do so. Same goes for data sent to the Arduino - unless you're specifically looking for it it will be ignored.
Opening the serial monitor does reset the Arduino though, so your sketch will be restarted as a consequence.
Are you suggesting the blinky LED sketch doesn't start when you run it from external power without the USB plugged in? If so, does the power on LED come on? What you're describing sounds to me to be a power issue with your wall wart.
Cheers ! Geoff