Simply put, Yes, but will require putting a bit of money into it, and possibly would cost around or a bit more than just buying one that's already like that.
You can't, since an electric scooter requires you to put a bigger and fatter rear wheel on the scooter, and chances are your push scooter is utilizing the skinny and rather flat plastic wheels. Also...
Yes but after buying all the parts and fabbing all the pieces it would be easier to just get a gas scooter.Unless you really need to do it then good luck.
If the wheel on the back of the scooter has one side free (the other side is taken by the belt drive to the electric motor), you can attach a gear to the free side of the wheel. From there, you can...