Non-trivial tractable properties of triples
I think your "non-trivial" property would need to be a property of sets of triples, not of triples themselves; i.e. you need that there are an infinite number of sets of triples that satisfy and do not satisfy the property.

Call a triple consists of 3 identical elements a 3-clone. Checking that a triple is a 3-clone can obviously be done in linear time, and checking that a set of triples are all 3-clones can also be done in linear time, because checking each triple is independent of every other triple. And there are an infinite number of 3-clones, and an infinite number of non-3-clone triples; so straightforwardly there are infinite numbers of sets of 3-clones and sets that aren't sets of 3-clones.

I think you need to strengthen the definition of non-triviality; any property of triples that is non-trivial can be lifted to a fairly boring non-trivial property of sets of triples, and there are lots of non-trivial properties of triples that are trivially easy to decide. Even the property of whether the first element of the triple is 1 works. I think the "interesting" properties of sets of triples are not just independently decidable for each triple and then combining the decisions. I'm not sure how to go about expressing that constraint in a rigorous way though.

As for a survey on the subject, I'm afraid I'd never heard of characterising NP-complete problems as properties of sets of triples, so I can't do much for you there.