Is this a genuine Omega Geneve or not?

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First of all, sorry if I didn't provide something required to make a thread.
I'm totally new to this forum.

My father got this watch at my grandfather's funeral and he gave it to me, but I don't know much about this watch. Can anyone spot a "fake" spot in this watch? (the engravings on the case back are names of my grandfather's relatives, I think)

For more information, all I know about this watch is:
1. it says it's automatic but manual winding does not work.(too "sticky" to turn the crown)
2. the automatic winding works fine and the second hand is much smoother than my Tissot Le Locle.
3. the "swiss made mark" says "-T SWISS MADE T-"(little space between the two words)
4. the fluorescent markings glow about half a second.
5. it has a day window and a date window.
6. the second notch of the crown moves the date(number), both clockwise and counter-clockwise.
7. the third notch of the crown moves the hands clockwise(forward) when turning counter-clockwise, and moves both the hands(backward) and the day(forward) when turning clockwise.
8. It makes a distinct ticking sound when changing the time by turning the crown.
9. Sorry for not providing the caliber details such as the serial numbers. I couldn't open the case back which was locked for many years...

the photos:
Edited:
 
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Welcome to Omega Forums!

The watch looks fine to me. The reference might be 166.0174 with caliber 1022 inside. A quick hand buff with polywatch or a similar product would clean up that haze on the crystal. I would get the watch serviced to fix the other issues you listed.