Why do two ends of a long conducting wire have the same electric potential? I am not seeing the "big picture" here. If I have two conducting spheres separated by a long conducting wire, why would the spheres share the same electric potential?
I think of the spheres as point charges, what does the conducting wire do? The $E$ field inside the conducting wire is 0, so what is it really doing?
 A: Because electrons flow when there is potential difference. So only when every point has the same potential will the system reach electrostatic equilibrium.
The above holds only if everything is conductor.
A: 
The E field inside the conducting wire is 0, so what is it really doing?

The potential difference between two points is related to the electric field along the path between them:
$$V_{ba}=\int_a^b \vec{E}\cdot{}\vec{dl}$$
So the fact that the high-conductivity material forces a (near-)zero electric field is exactly why the two ends of the conductor are at the same potential.
As other answers have said, the reason why the E field is zero inside a perfect conductor is because if it wasn't, current would flow until it became zero; and electro-statics only considers situations where all such flows have reached steady state.
A: Think of a gas. We will ignore gravitational potential, and we'll consider the situation to be at steady state. Compressing the gas takes energy, so we conclude that the entire container of the gas will be at a constant level of compression - in this case, constant density as well. You may intuitively understand that this is true no matter how oblong the shape of the container is. This container could be a compressed air pipe running a mile. The two ends still have the same densities because that's just how gases work.
Electrons in a conductor are like a gas which occupies the same space, and is confined by, the conducting material. The only critical difference between the electron gas and a classical gas is the existence of long-range forces. The electron gas is charged so one end affects the other by the 1/r^2 electrostatic repulsion. Otherwise, the gas still takes some energy to compress (on the Debye scale), but its distribution is mostly dictated by minimizing the electric potential.
Hope that analogy helps your understanding some.
A: Opposite charges attract, electrons are moving charges. A conducting wire is the perfect way for electrons to move until the charge of the two ends is equal, and this happens very quickly.
A: See.  If there is some potential difference between the 2 ends of a wire the electrons will flow in a direction where there is less potential difference.  Hence the potential difference will be uniform throughout the conductor.
A: The two spheres share the same potential if and only if they are in equilibrium, which in general, as you pointed out, does not have to always hold.
In fact, if you take any two spheres and connect them with a wire, they will not share the same potential. They eventually will when all the charge carriers have flown from one to another according to the direction of the electric field therein.
A: The electric potential in a place is the potential energy per unit charge that would be placed in that position. So, for a particular charge, a difference in electric potential between two places, means a difference in potential energy in those two places.
Now, as everything else in this universe, a charged particle tends to minimize it's potential energy when you lift the restrictions. To make this clear, stand up and take something not to valuable in your hands. Now watch what happens with that object when you remove the restrictions (your hands): it lowers its potential energy as much as possible.
Now back to the conducting spheres. Let's assume they start at a different potential. That means free electrons on one of the spheres can lower their potential energy by moving to the other sphere. However, there are restrictions: the electrons can't travel through the air between the spheres. This restriction is lifted when they are connected with a conducting wire. Free electrons move from one sphere to the other. As this happens, the potential in both spheres changes as well. Free electrons will continue to flow from one to the other as long as the potentials are different, but eventually, they become the same. Now, there is no longer a difference in potential and the charges stop moving. When this equilibrium is reached, and only then, the potentials are the same.
A: When we connect two conductors through a conducting wire then the new system we get also behave as a one new conductor, so as we know in a conductor potential is constant everywhere , we can also take this in this way and also we do not assume any charge on wire due to fact that charge on surface is inversely proportional to radius of that surface
A: I am supposing that by point sphere you mean to say that the radius of each sphere is much smaller than the distance between these spheres so that charge in one doesn't effect the other directly.
By definition conductors in electrostatics have zero net Electric field inside.From this it directly follows that Voltage has to be the same everywhere in the conductor body
Now let me give a physical explanation of why this happens:

Conductor is a property of materials where the charged particles are
free to move around within the conducting body.

This together with Coulombs Law that states that like charges repel and electrostatics (where nothing changes with time) means that these charged particles in the conductor should each have Zero net Force on them (otherwise force will lead to an acceleration and hence change of position with time, that violates electro'statics').
Hence charged particle in your system of two spherical conductor joined by a conducting rod are equally distributed between the spheres and further they are repelled by each other(coulombs repulsion) and has no choice but to stay at the very surface of the conductor (because that's the extreme charged particles can go, they cant jump out of the conductor).
Hence these charges in a conductor rearrange almost immediately into a position where the net Electric field (force) on each individual charges are zero.
