Inglis of Canada made MANY Browning HPs. Mainly there are two versions. One like your's and the other with a fixed rear sight.
The matching numbers is good, as that's the way they were marked (i.e. on all three pieces).
The "*" in MkI* indicates it has an improved extractor over the MkI, but most are this version, so no biggie there.
The stock may be original or a repro. Either way, it's leagal to use on your BHP as long as it's the same design as the original (which it is according to your description).
I don't know about the yards/meters on the sight. Trying to shoot someone at 500 "x", I honestly don't think it's going to matter much in accuracy if it's yards or meters. If you can hold it that tight, you can adjust for the slightly longer meter/shorter yard.
The Chinese bought a huge bunch of these exact guns. Many of these were imported into the US back in the 80's. I had one. I don't think your's is one of these though. IIRC the Chinese guns had some prfix on the S/N like "CH" or something. My current Inlis (fixed sight) has a "T" prefix and honestly I don't know the significance of that either. If your's has no prefix, it may be worth quite a bit more than a Chinese contract gun would be. Definately check it out in detail with a BHP expert before selling it.
Mine didn't shoot any better with the shoulder stock on or not. I would have thought it would have made a difference, maybe it does at longer range. Mine fitted rock solid, with no movement, but it was a repro I bought and fitted myself. The shoulder stock isn't all that useful in reality, but it's WAY COOL no matter what anyone says about utility.
As for the other questions:
It's safe as any gun to shoot with standard ammo. The Brits still use BHPs and that's with NATO ammo. I'd lay off the +p or NATO spec stuff just because the possible value of the gun makes no sense to wear on it when you don't have to. I'm no great fan of +p in anything anyway, so there you go. But normal, standard 9mm should be fine.
The button on the rear sight should push in IIRC. I could be wrong as all of this is on memory from way back, but it's just like the tangent sight on a KAR-98 or similar. The button locks the rear sight in place at the range selected. There is no windage adjustment, other than drifting the sight, same as a fixed sight model.
13 rd BHP mags will fit and function fine. As well as they would in an Argentine, Belgian, etc. They are interchangable.
It may have problems with JHPs. Most older guns do. Try it and find out. That's about the only way. It shouldn't have problems with FMJ.
As I said, there may very well be some collector value to the gun. Check with someone who really knows BHPs, like the forum cited above.
Ross