Is there a way to test "Turing completeness"?
I don't know about specific models. It's pretty easy to become Turing Complete, since all you need is infinite search. So I can imagine a model where searching for finite vs. infiniteness boils down to Turing Completeness, but I don't know if any actually exist.

But a general algorithm "look at an arbitrary systems and tell if it's Turing Complete" can't exist. Otherwise we could do this:

RunTuringMachine = lambda (M1) . lambda (x). lambda (M2) . 
  run M1 on x and discard the result
  run M2 //we only get to this line if M1 halts on x


Then, for any inputs M1, x, RunTuringMachine M1 x is a Turing-complete system if and only if M1 halts on input x, and we've solved the halting problem.

We could also prove this by Rice's theorem: if we can decide if any system is Turing Complete, then we can decide of a Turing Machine is a Universal Turing Machine, which is impossible by Rice's Theorem.