While using arduino to close a circuit (the output to transistor's base), is it necessary to connect negative of circuit to ground in arduino?
You will need to ground the emitter/DC jack's ground with the Arduino. Without doing this the base has no reference point to be driven from.

As for the DC socket. Check first which contact of the plug is positive and which is ground. Most often the internal contact is positive, but not always. Once you've done this, either check the datasheet of the socket or use a multimeter in continuity mode to check which pin the middle is connected to. MY guess would be the one on the back is connected to the center pin, and the other two will be the outer.