Do battery electrons only move if there is a positive terminal at the end of the wire?

Electrons will flow as long as there is an electric field to move the. When you first connect the wire to the negative terminal of the battery, the electric field generated by all of those electrons stuck on the negative terminal will cause them to move into the wire. They will basically move to distribute the electrons evenly along the wire. This happens fast. Really fast. The exact speed depends greatly on your particular wire, but we're talking microseconds, even over 1km of wire.

Now eventually the electrons reach the far end of the wire, where they get to interact with whatever you hooked it up to. If this is the positive terminal of the battery, then there's a bunch of positively charged molecules that they can combine with to become neutral. This is basically always a desirable thing, from an energy perspective, so they do it really fast. Once those molecules are neutralized, the chemical reaction in the battery is knocked out of equilibrium, and it starts generating more electrons at the negative side and more positive ions on the positive side. It does so by depleting chemical energy. This opens the door for more electrons to flow through the wire, and the expected result occurs: a short circuit.

Your third scenario basically never happens in reality to any meaningful degree (outside of exotic things like Van de Graaff generators... and scuffing your feet on the carpet). What ends up happening is an electrostatics problem. You keep plucking electrons off of the end of the wire, so the electrons keep re-distributing. As they do so, the entire structure (wire and batteries) becomes more positively charged.

Now how did you pluck the electrons off the wire in the first place? You had something that was more positively charged, so that the electrons wanted to go in that direction. Short of tiny tweezers, that's really the only way to pull the electrons off. But now your whole object is becoming more positively charged. This diminishes the electric field you were relying on to pull the charges off the wire. Eventually you reach an equilibrium where the charge on the wire is exactly right to have no electric field between the end of the wire and your device that's pulling electrons. The flow stops there. (and, incidentally, the positive side of the battery is ever so slightly higher in voltage. Whatever charge the negative side had to reach equilibrium, the positive side has that plus the EMF of the battery).

Now keep this going to an extreme, and weird things start to happen. If your object gets positively charged enough, you'll eventually reach the ionization potential of the air. When this happens, electrons will flow through the air onto your object, creating an arc. This is exactly what is happening when you charge up a Van de Graaff generator (except in reverse. Typically those generators create a highly negative object... but the same rules apply).

Now your battery and wire has become "battery, wire and walls" (and possibly you--did you remember to leave the room before engaging in high voltage activity?). This means you have more electrons to distribute. Eventually you will swamp whatever positive source is drawing all of those electrons away.

If you want to see what happens in this situation, check out the Dueling Tesla Coil Dudes. They have some generators configured so that they take a bunch of electrons from one guy and move them to the other. Most of the time they just interact with the air around them, neutralizing their charges. But when they get close enough, it's time to pay the piper!

Electrons are actually always moving, even when there's no potential difference (aka voltage). They just move in random directions.

What happens when there is a potential difference, i.e. when the wire is connected to two terminals of the battery, is that although the electrons continue to move in random directions, there's now a slight bias towards one direction. That's what constitutes the current. The speed due to current is actually very slow, usually a few millimeters per second. This is in contrast to their random speed which is much faster. See the Wikipedia article on drift velocity.

If only one end of the wire is hooked up to the battery, then there is no potential difference, and no current. You can't discharge a battery by connecting only one terminal to "something" either. That something will just become charged too.