Inductor potential difference direction If you have increasing current going down an inductor, the inductor will produce a potential difference to produce a counter current in the opposite direction, or up. The lower potential is then on the top of the inductor. However, the net current is still going down, which is higher potential. How can net current go from lower to higher potential?
 A: Your initial supposition, "If you have increasing current going down an inductor,..." is a bit vague about exactly what external influences are at work on the inductor in question.  I will suppose that the current through the inductor is increasing because a battery has just been connected directly to the inductor.  Other suppositions are possible, but this is a very simple explanation for the increasing current, and the concepts at play would be similar in any other more complicated circuit.
At the moment the battery is connected to the inductor, no current has been flowing through the inductor, so it has been producing no potential difference of its own.  And the battery is applying a higher voltage to the top of the inductor than the bottom.  So current will begin to increase in the downward direction.  How quickly is it increasing?  As quickly as the inductor will allow it to increase, because the more quickly the current is increasing, the more counter-voltage the inductor produces, and the less voltage there is to push the current.  Specifically, the current increases at precisely the rate for which the inductor voltage cancels out the battery voltage exactly, allowing all of the classic circuit diagram laws to be satisfied.
We do not grow up experiencing inductors directly on an everyday basis, so to intuitively understand their behavior, an analogy (https://en.wikipedia.org/wiki/Impedance_analogy) to mechanics is often useful:
Inductors resist changes in the current passing through them, just as a large mass resists changes in its velocity.  Your question is analagous to asking how an object being accelerated by an external force can ever move, since it always produces an equal and opposite reaction force opposing its direction of acceleration.
A: The counter potential difference you are talking about is indeed produced but it is mostly smaller than the dominant potential difference (such as potential different due to a battery) due to which current was increasing and hence the net current will still flow in the same direction because the initial source of potential difference overcomes the opposing potential difference but definitely the current is reduced than it would have been. It's is based on Lenz's.
A: the inductor has an electrostatic field and a non-electrostatic field inside its metal, which balance and cause a zero electric field in the metal of the inductor. It's zero since the inductor is assumed to have approximately zero resistance, so for the current density in the equation J= conductivity . E to have a finite value, the electric field must go to zero, since the conductivity goes to infinity. Now, where did the potential difference come from? Remember that a potential difference is defined for electrostatic fields, and because the two fields balance, the potential difference is going to be the opposite direction of the induced electromotive force and same magnitude.
A: If you have increasing current going down an inductor, the inductor will have an induced electric field going up to resist the current increase. Since a perfect conductor has 0 net electric field, the electrostatic charges in the circuit will rearrange such that the electrostatic electric field points down and has equal magnitude to the induced electric field.
The voltage potential only accounts for the electrostatic electric field. Hence since the electrostatic electric field points down the bottom of the inductor has a lower voltage than the top.
Hence the OP's statement that the bottom of the inductor has a higher voltage is incorrect, and the current is still going from high to low voltage.
