What exactly happens when we increase the magnetic flux? According to Lenz's Law, the current would try to counteract the change in magnetic flux, but what happens after current produces counteracting flux? Since the flux would decrease would we get a current flowing in the opposite direction which would then cause the flux to increase causing the current to change again? Will this Loop ever stop? If yes, What would be the flux at the end? Would it be equal to the flux before the experiment?
 A: Just because the current opposes the change in flux does not mean it will completely counteract that change. Think about an LR circuit that is closed at t=0. Without the inductor, we would just have a resistor hooked up to a battery and the current would flow from the positive to the negative terminals of the battery. In our LR circuit the inductor opposes this, but eventually the current will stop changing and it is as if the inductor is no longer there.
So to answer your question, there is not a loop that forms. The opposing current does not reverse the change in flux, it just opposes it. It is similar to how inertia causes objects to oppose changes in motion. It does not mean that mass causes objects to move in opposite directions to applied forces.
A: Did you try to imagine why a permanent magnet, falling inside an aluminum tube, gets slowed down? The changing magnetic flux of the falling magnet induces eddy currents in the tube, this currents induce a magnetic field in opposite direction to the permanent magnet and this slows down the falling magnet. Slower falling is accompanied by smaller flux and at the end there is a equilibration between the mechanical force, which moves the magnet and the magnetic force, that slows down the magnet.
Sometimes it’s a good idea to see what happens on the atomic level. To understand the phenomenon of electro-magnetic induction one has to remember that subatomic particles obey a magnetic dipole moment and that moving electrons (protons) under the influence of an external magnetic field get deflected (in a direction, perpendicular to both the direction of motion and and the direction of the magnetic field (Lorentz force). Furthermore one has to recognize that the relative motion of the involved participants is important only, it doesn’t play a role, will the charge moves or will the flux changes!
With this knowledge one could imagine how the changing flux of a moving magnet induces in a metallic tube a current of moving electrons. This curvilinear moving electrons have aligned magnetic dipole moments and produce a common magnetic field, which is in opposition to the permanent magnet.

According to Lenz's Law, the current would try to counteract the change in magnetic flux, but what happens after current produces counteracting flux?

As told above it comes to an equilibration, the counteracting flux always will have a partially strength of the inducing flux.

Since the flux would decrease would we get a current flowing in the opposite direction which would then cause the flux to increase causing the current to change again?

That is what happens every time one opens or closes a current loop. Even changing the Potential difference in a loop it happens.

Will this Loop ever stop?

Every physical process is accompanied by loss processes. So every change in current - in addition to “sideway” effects like zig-zagging of the electrons inside metallic lattices - accelerates the electrons and lead to EM radiation. Alternating currents are struggeling with this problem all the time, electric power transportation with direct current is less energy posting.
A: 
Since the flux would decrease

Why would the flux decrease?  Assume for clarity, that there is (small) loop of ideal conductor through which there is zero current and, further, there is zero external magnetic flux threading the loop.  It follows that, since there is no loss in the ideal conductor, the total magnetic flux threading the loop must remain zero.
In other words, any attempt to thread the loop with an external flux will result in a current through the loop producing a magnetic field that precisely maintains (at all times) the initial condition of zero total magnetic flux threading the loop.
For a non-ideal conductor (non-zero dissipation), the perfect cancellation of external magnetic flux threading the loop no longer holds.  Do you see why this is?
