How can I control distortion in an op-amp circuit?
Are there inherent non-linear/distortion characteristics of an op-amp?

Yes.  Bare opamps are actually quite non-linear, or at least their linearity can't be counted on much.

Or is this all contingent on implementation?

No, since you're asking about opamps, not the circuits around them.  The opamp is still a opamp regardless of what you put around it.

What are the tried and true techniques for minimizing distortion in an op-amp?

I don't know since I'm not a chip designer.

Minimizing distortion in the overall circuit that includes a opamp is usually done with negative feedback.

Added

You have now completely changed your question, asking about opamp circuits instead of the opamps directly.  That really should be a separate question, but since I'm here, I'll address it at least a little.

As I mentioned briefly above, quite linear circuits are created with opamps by using negative feedback around them.  The very high gain of opamps contributes to making them non-linear in the first place, but it also allows for feedback to get good linearity at modest gains.

I go into the math behind negative feedback in detail here.