Try doing some weights also. I think that may help you burn fat a little after you leave the gym. Good luck.
Yes you are. Running is fine.
The objective of losing weight is to be active. Obviously, if you're just starting, start slow. No one just randomly says "today I'm going to be a pilot" then takes off in a jet airplane. Baby step is the sure and safest way.
To be fit and healthy is a lifestyle. Although you may not see results in a couple of days or even weeks but if you keep at it you WILL see drastic changes.