Is it decidable whether a Turing Machine will visit every non-end state from input x?
It's undecidable because you could use it to decide halting.

Because the number of states in a Turing machine is finite you can enumerate all $2^n$ combinations of possible non-end states. Run the algorithm against all of them. 

If the algorithm returns yes for a group of states that includes an original end state then the machine halts.

Halting problem is undecidable so the algorithm cannot be executed by a Turing machine.