Circuit understanding
Fairly close...


The chemical reactions in the battery create a difference in potential between the two terminals, whether there's a conductor hooked up or not. Connect the conductor, and that potential pushes charge through it (allowing the chemical reactions to proceed faster, like a fire when you allow air in freely)
Exactly : that potential difference creates the electric field which exerts force on the charge.
Again, correct : that unit of charge ( = 1 Coulomb = 1 amp for 1 second) carries 9J from 9V, or 18J from 18V.
Correct twice again : unless the conductor's resistance R changes, twice the voltage pushes twice the current; AND each Coulomb carries twice the energy. This means 4x the power; which is expressed as P = V^2/R.


(Note that the conductor's resistance may change : in a lampbulb it increases as the wire heats up, so the current won't double; it may also become infinite if the conductor is a fuse!)