Like all statements, strictly speaking I can only interpret “infinitely many” in the praxeological sense of guidance for my actions. So it would render to something like, “I’m sure I will never run out of twin primes no matter how high I look.” My answer to the first question is then “it seems likely.”
I can only interpret “true” praxeologically, so yes, either the belief “I’m sure I will never run out of twin primes no matter how high I look” pays rent in guiding my actions, or it doesn’t.
“Either P or not P is provable” in this context can only mean "the axioms of arithmetic can be semantically transformed in a standard way (based on English grammar if loosely worded, or based on the rules of logic if written formally) to yield either:
“I’m sure I will never run out of twin primes no matter how high I look.”
“I’m not sure I will never run out of twin primes no matter how high I look.”
Whether or not the axioms can tell me which of these is the case, it is patently obvious - and I need no law to be sure - that one of them has to be the case. Either I am sure or I am not sure. It doesn’t take any principles or laws to make this clear as long as we understand that all beliefs and all statements are only interpretable as praxeological ones, that is, as guideposts for one’s action. It is only the fact that many statements appear to be “objective” that people ever need to be reminded of the Law of the Excluded Middle (and the others). I’d be surprised if anyone seriously held that they can be neither “sure” nor “not sure” of something at the same time (though word confusion makes anything possible).