How does electricity flow in conductor when potential difference is applied?

Electrons will flow against the electric field lines because their charge is negative, and the electric field thus exerts a force $\mathbf{F}=q\mathbf{E}$ on them which is in the opposite direction. Thus electric field lines inside the wire go from the positive to the negative terminal and the electron flow goes from the negative to the positive terminal. Electric current goes, consistently with both of the above (because the electron charge is negative), from the positive to the negative terminal.

The electric field lines will twist with the conductor if you bend it into some weird shape. (This is due to slight charge buildups on the wire bends and is beautifully explained by Purcell.) For the situation you describe, the electric field lines and the wire pretty much match already so just draw some more lines. You've already explained current flow in terms of electrostatics in a circuit like this! the only snag is what the state of affairs is inside the battery, but that's another story.

The wire sets up charges on the surface to channel the electric field exactly along the path of the wire. This isn't surprising, if the electric field doesn't exactly follow the path of the wire, charges are shunted to the surface, and these charges will then move the electric field so that it is parallel to the wire. The amount of charge required to do this shunting is tiny, it's a negligible capacitance of the wire that depends in a crazy nonlocal way on the shape of the wire, the type of battery, and the other conductors around.

But the answer is just yes: the metal conducts charges to the surface just until the electric field is going parallel to the wire along the entire length of the wire, no matter how many times it doubles back.