I would guess an early fifities (3) watch and with 2576 as the ref. (1) - but you need to have it opened to be absolutely sure and at the same time you will get the movement caliber (1). and the movement serial - getting closer about the manufacturing year (3). The crown is not original (2) - as you suspected but a correct replacement is available. It will look like this:
Bracelet is not original (2) either and I would recommend you to get a quality 18mm (at the watch end) leather strap instead.
(4) Yes.
It is not very valuable but case, dial and hands look very nice and you will get you money back. I do not know if your mother has given you the watch or if you are supposed to pay for it? If given - it is a no brainer. If you have to pay - it depends of course on what sum...
Be sure to tell the watchmaker not to polish the case or try to clean the dial. Get the movement serviced, an new Omega replacement crystal and crown and you are set. If you like it and want to wear it - you have to get it serviced as - if you start to use it/wear it without - you might damage the movement (it is very probably running dry on lubrication).
Good luck - that is a very good and nice starting point in Omega collecting.
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