To reiterate: because these programs both loop infinitely and perform I/O (with the I/O data not being computable from anything the program has when it begins running, not being "predictable" in any sense by the program), they are physically-realizable programs that simply don't match the neat analogy of normal Turing machines.

They match the neat analogy of Turing machines that start with possibly infinitely many non-blank squares on their tapes. All the obvious things you might like to know are still undecidable. Is this machine guaranteed to eventually read every item of its input? Is there any input that will drive the machine into a certain one of its states? Will the machine ever write a given finite string? Will it eventually have written every finite prefix of a given infinite string? Will it ever halt? These are all straightforwardly reducible to the standard halting problem.

Or perhaps you would add to the model an environment that responds to what the machine does. In that case, represent the environment by an oracle (not necessarily a computable one), and define some turn-taking mechanism for the machine to ask questions and receive answers. All of the interesting questions remain undecidable.