Sorry not all the photo loaded
I'm very sorry for the loss of your father, and I see it is so recent as well, so you must still be feeling it. It was lovely to read your story, but to lose a parent too young is even more sad than it would otherwise be.
I have spent a pleasurable few minutes analysing your watch against the book Flightmaster Only, and if you are at all interested in the history of the watch model, I really recommend it.
Dial - I suspect the dial is original. It is marked T Swiss Made T. Service dials are marked "Swiss Made" only. The dial dates your watch between 1972 - 1976 (not having any dots at the 3, 6 and 9 positions). That's why the lume is poor - the tritium paint has lost its luminescence.
Hands - I suspect the yellow hands are original, mainly on the basis that the main second hand appears to be of the thinner and the original type. The white hands look like replacement hands filled with luminova; I think this because of the shape of the end of the hour hand and how white they are. A way to check is to put the watch in the dark after wearing in daylight - if the luminescence persists beyond a few seconds, they are likely replacement hands (but correct). On the yellow hands - about one third of the production had yellow hands, which were supposed to be more visible in cockpits with red lighting. It would be interesting to get a look at the GMT hand, which is obscured in your photos.
Bezel - looks original, or at least an original part. The current replacement part has serifs on the numerals.
Bracelet - you are right, the correct reference bracelet for this model is 1162/172. The 173 was for Speedmaster MkII variants. I think it is 2mm thinner in the endlinks, to fit 20 mm lugs instead of 22 mm. You can still find the correct bracelet.
Pushers - Yes, these are not the correct pushers, nor the correct crowns. Replacement service parts for all these are still available. The main crown is not correct either - it should have 22 teeth.
If this was my watch, and I wanted to wear it daily, I would go about it in this way. I would find a trusted watchmaker with an Omega parts account. I would not send it back to Omega as they may, as you suspected might already have happened, replace the dial. I would have it properly serviced. I would have all of the crowns and pushers replaced, and I would also have the mineral crystal and the seal replaced. I would do that because the pushers and the seals all affect water resistance. I appreciate you won't be going swimming in it, nor can the water resistance of such an old timepiece be guaranteed, but having new seals can't hurt. Get all the old parts returned to you, and you have both retained the history and maintained the watch to be a working faithful original.
I would not get the case refinished, given its sentimental history, but it could be done.
Hope that all helps answer your questions, and enjoy the memories the watch gives you.