Lenz's law and the magnitude of opposing fields
I think you may be struggling conceptually here so I'll clear a few things up first. If you were driving a regular transformer with AC there would be three currents of interest: -


The current that flows in the primary when no load is on the secondary
The current that flows in the primary due to secondary current
The secondary current (due to a secondary load).


What you find is that current in the primary with no load is simply due to the inductance of the primary and the voltage applied - it is often called the magnetization current and is the same as any current in an inductor with a voltage applied across it - it generates a magnetic field and sometimes there are eddy current losses and sometimes the core can saturate a little bit.

It generates the magnetic field that induces a voltage on the secondary as per faraday's law of induction. So, the secondary gets an induced voltage and, if you connect a load a current flows. There is also an exact opposite current (if it were a 1:1 transformer) that flows in the primary due to the secondary load.

Those two load currents would (if you could untange them) produce exactly opposite magnetic fluxes and the only flux that remains is the same old magnetization flux and this ensures (quite nicely) that the induced voltage on the secondary is proportional to primary voltage and turns ratio no matter what the load (within reason and in a fairly lossless transformer).


  When current flows through the primary coil, a magnetic field is
  produced in the core, which induces a voltage in the secondary.


It is the rate of change of magnetic field that induces a voltage in the secondary not just the presense of a mag field.


  I feel like Lenz's law says that the secondary coil produces a
  magnetic field in the core the opposite direction of the field
  produced by the primary, but I'm sure it can't be the same magnitude.


Yes it does (as per my words above). Actually, the ampere turns in the secondary due to load current are exactly opposite to the primary ampere turns due to that secondary current. 


  What determines the magnitude of the magnetic field produced by the
  secondary coil?


You can't really measure it because it is cancelled by the load induced magnetic field in the primary i.e. it neither adds to or subtracts from magnetization current described above.

As for your middle paragraph I'm unsure what you are driving at: -


  Suppose I have a step down transformer where power is put through the
  primary coil in one direction only, then stopped, then repeated, at a
  frequency of 60 Hz.


When the process of applying power is halted a DC current will flow that induces no secondary voltage and takes the core towards saturation. Step and repeat the process and you'll get core saturation and problems. Transformers do not pass DC - the average voltage applied to the primary is ideally zero and the secondary rewards you (hopefully) with an average output voltage of zero.