1) I didn't say we were, but when someone is called reasonable or acting with reason, then that implies deterministic/algorithmic thinking. When we're not deterministic, we're not reasonable.
2) Yes, to reason does imply to be infallible. The deterministic algorithms we follow are usually flawed.
3) I can't speak much to that, but I speculate that if "AI" can do reasoning, it would be a much more complex construct that uses LLMs (among other tools) as tools and variables like we do.
* Humans that make mistakes are still considered to be reasoning.
* Deterministic algorithms have limitations, like Goedel incompleteness, which humans seem able to overcome, so presumably, we expect reasoning to also be able to overcome such challenges.