FF, I'm not 100% sure of this, but I believe the time signal is uploaded to a series (?) of satellites and broadcast from there. I do know there are a number of government and research facilities that do the same thing, i.e., Lawrence Livermore Labs, Oak Ridge, etc.
#2 I can answer. Other than GPS, all hand held altimeters are based on barometric pressure, and yes, there are a lot of variables.
What you need to do, each day, is reset to a known altitude. For example, I know that the ranch office is at exactly 1240', as determined by a 7.5 minute USGS Topo. I'll reset to that just before I go out on trail (not that I really need it in this area).
Rapidly moving weather fronts will affect it somewhat, but at least in my experience, not all that much. The most deviation I've experienced is about 40', but understand, this wasn't at high altitudes like, say, 8000'. It's not precision, but it's usable.
#3, no, you can't adjust for declination, but I've never found that to be a problem. Each year, I check with a surveyor friend for that year's magnetic declination. It does vary a little bit from year to year. Knowing that, I just take a reading and apply a little "Kentucky windage" to get true North. Again, it's not precision, but it's usable.
I don't think this watch was ever meant to be a serious instrument, but it does give you useful info in a pinch. If you're going to do some serious trekking in the Montana backcountry, you probably should be packing a GPS, along with topo maps, a good compass and altimeter as a backup.
TFL Members are ambassadors to the world for firearm owners. What kind of ambassador does your post make you?
I train in earnest, to do the things that I pray in earnest, I'll never have to do.