Newb questions about my vintage 30T2 RG Chronometre (need NY Watchmaker)

Hi Folks,

long time reader, first time post-er. I just inherited a fun little watch that belonged to my girlfriend's grandfather before he passed away(pics below).

The dial is pretty weathered but I love how it looks and it's really special to me for obvious sentimental reasons. Kind of excited about the chronometre grade movement as well, although I don't think it's anything special as far as I can tell(I think it's a 30t2sc movement from the 50's?). Just ordered a new strap and very much looking forward to wearing it.

I had a couple of questions:

-How often does one wind a mechanical watch like this, and how do you keep from over-winding? I've only owned automatic watches until now.

-Unfortunately I forgot to take a photo of the case back, so perhaps more information is needed, but is there anything else you could tell me about this watch? For what it's worth, I really liked this little series on 30T chronometers: https://squeletteblog.wordpress.com/2016/01/18/part-iii-chronometers-30t2rg-and-30t2scrg-english/

-Should I have this watch cleaned, polished or serviced? The watch wasn't working when I originally found it. We brought it to one of those awesome old dudes in Hong Kong who looks like they've been in business for a hundred years fixing watches out of a cupboard on the side of the street and he got it running again. I'm not entirely sure that it's in perfect working order, I've found it stopped several times although that could be due to me under winding it(I'm a little paranoid about wrecking the movement). Now that I know it works, I was thinking of maybe bringing it to Grand Central Watch Repair(I live in NYC) and having it thoroughly serviced. Any tips or advice of what to have done and where to take it?

Anyways, thanks for your time. Apologies for any unwitting breaches of forum etiquette!

Also, that bracelet looks valuable in its own, if it's a original to the watch, and if it's an omega, even if not(not sure if omega was doing their own Milanese ones at this age) then I would deffinetly keep it with the watch. I would wear it but It's smart to wear it on a strap to preserve the bracelet. Don't just send it to anybody, make sure they are competent with these old omega's.

Like cicindela has said it is very valuable due to its unmolested original state.

Thanks for all the feedback! If anyone's got a recommendation for somewhere in NYC that can service this properly, it would be greatly appreciated.

Please correct me if I am wrong. I think it is a 33mm case (no spacer ring) and there doesn't look to be 4.5mm of case surrounding the movement. The 2367 I have is 35.5mm and you can see how much more steel is around the movement. As I said if I have this all wrong please correct/forgive me and I will remove this post.
OP's

Mine

Wow!! You're a very lucky man! If you want to make a trip out to Jersey (Bergen County) I know a good watchmaker where I got my SC RG serviced. Send me a message if interested.

Thanks so much for all the feedback and info, very much appreciated. Pic below with the a new Hodinkee taupe strap.

I ended up taking it to Grand Central Watch for a quick look and they seemed to think it was running well(I was under-winding it -_- ) and that no service was necessary in the short term. I'd still like to have a qualified someone give it a once over and clean the movement, but I think I'm just going to wear it for a couple of months first.

Still looking for some NYC watch repair recommends if anyone has them, Central watch seemed pretty expensive although I'm not opposed to paying their fees if that's the best option.

Anyhow, thanks again.