GPS systems can, and do, work everywhere you can get satellites. The coordinate system -- or datum -- you choose to use should vary (even in the US) depending on the maps you are working with, your project needs, the area you are working in, or what the legacy system you are dealing with...
In the history of map-making, a ton of different projections and datum have been used to try and describe spherical space on a two-dimensional map. Things like NAD27 simply refer to a certain system (North American Datum 1927) and if a map you are using was created with NAD27, then points recorded in NAD83 datum won't line up as expected.
Confusing? It is. Especially once you get into the details. But for most of us, it doesn't need to be. At the most basic, each GPS stores your position in one consistent datum. Garmin (and most if not all other GPS systems) uses WGS-84 (the World Geodedic System - based on the gravitational center of the earth). Even if you set your unit to display in Lat-Long or others, the unit itself does internal calculations to give you that readout - but the position itself is permanently stored as WGS-84.
As for your compass - the only difference (as far as I know) between a Northern and Southern hemisphere compass is how the needle is balanced. Since the magnetic field pulls downward in the Northern Hemisphere, the needle is counter-weighted to keep it flat when the compass is held level. In the southern hemisphere, the change in angle can result in the compass needle dragging - and throwing off your accuracy (though it will still point in the right direction (assuming you have your declination set correctly for your region.))