This is a most fascinating thread. Since I have been reading, but not commenting, my comments are responses to several different scattered items here (not all in order...):
1. He thinks you are a firefighter because you understand him -- In certain professions, only another in the field can truly relate a person's experiences. Clearly firefighting is one of those. Also as a nurse, you may be the only person he sees regularly who reminds him of his world and the types of people with whom he interacted frequently.
2. You mentioned that he doesn't like it when people call him "Fireman [his name]". The likely cause of his objection is the name. When one is reincarnated after generally liking a past life that was not long enough, it is a setback. He was captian or leiutenant, accomplished in his field; now he's a kid. Having his professional status reduced to "Fireman [kid's name] is a severe, though unintentional, insult.
3. Related to item #2, when you mentioned near the beginning of this thread that he insists on Fire/Rescue, instead of just firefighter, that was my clue to spend time reading more. That distinction would only matter to one in that field, and there is almost no way at all that a young child (unless raised in a family of a fireman/rescuer) would ever learn to be particular about this. It would not be in a kid's story, nor taught in school, nor in a TV show or movie.
4. Your firedrill at home is also most interesting... covering with a blanket to run through, and getting low are key techniques that would be used in a real fire emergency. His instincts were probably more advanced than yours. If you ever have a fire, he'll know what to do.
5. You asked whether you should teach him, or he should teach you in the pool. Do both! Let him lead, and remember he is an intelligent person with something to bring to the project, but help him when he doesn't know something, and maintain enough control to ensure safety without being overprotective. That's a tall order, I know, but it a suggestion of the ideal. It will let him build on what he already knows from the past and may help him feel better about being able to do real things (not just play and toys) in this life.
6. I must confess to a brief bout of goosebumps when you mentioned "pages". I do have a memory of seeing my experience as a large book. Images of current life were near the end of the printed portion. Turning back only about what would have seemed a chapter's worth of it brought me near the beginning of this life. Then I turned back many more pages and saw pictures of my previous lifetime. It all occured in a state between sleeping and waking somewhere. I felt as if at that moment I had access to the whole story, but then awoke before having time to see more.
7. Thank you again for letting us know about this most interesting case, and for trusting our community for answers. My one suggestion is to take him seriously, and explain calmly that he is here with you and safe now. If he maintains his interest, let him (without pushing him) study and see more advanced books, movies, etc. Perhaps he will be a great rescuer (again!) some day.
8. I seem to remember, but am not sure, that one or more of the companies that responded during 9/11 was using recently upgraded equipment (new truck). You may want research that if you are working on finding his identity -- which I do recommend doing.
...Rod