Solenoid and mechanical damping, options for ways to minimize
Ok, I'll take a crack at this.

No matter what you do, this isn't going to vibrate continuously. Since what you've created is essentially a spring/mass system, where the "spring" includes a magnet. I'm not sure if that's your intention. When you increase the coil current you are making the "spring" stronger and increasing the resonant frequency.

I believe your explanation of the eddy currents is correct. The iron rod is creating eddy currents as it moves through the stationary magnetic field of the coil. Making the rod non conductive, or reducing the conductive loop diameters (by making cuts), can mitigate this.

I believe the back-emf will only be created as the iron bar becomes magnetized (which it probably will). Once that happens, it will start acting like a generator opposing the flow of current into the coil.

The key here is that the back-emf is a voltage source. So if you're driving the coil with a voltage source the current will be reduced.

Current = (voltage - back-emf-voltage)/resistance

Note that the field created by a coil is proportional to the current. It doesn't care what the voltage is.

You can avoid this problem by using a current source instead of a voltage source. The back emf will reduce the apparent voltage, but the supply will simply increase it's voltage to compensate, thereby keeping the current, and magnetic field constant.

The "large resistor" solution, is basically a crude way to create a current source. This works great for low power devices like leds, but as you've noted it would be quite wasteful for something like this.