Whether a Turing Machine ever halts is a concrete question with a definite yes or no answer
This doesn’t seem right. In different models, different TMs halt, and it’s impossible to specify which model you really mean as settling the question. You can even decide, of your own free will, whether certain TMs halt or not (if you’re simulated by them, say). So, it should rather be the more trivial “In any given structure, whether a TM execution defined on it halts, is a property of that structure.”
This doesn’t seem right. In different models, different TMs halt, and it’s impossible to specify which model you really mean as settling the question. You can even decide, of your own free will, whether certain TMs halt or not (if you’re simulated by them, say). So, it should rather be the more trivial “In any given structure, whether a TM execution defined on it halts, is a property of that structure.”