Hi,
Yes, you have a lot of questions that have long answers! First, have you tried reading the map projections guide book in the help? There are several topics each on geographic and projected coordinate systems.
Geographic coordinate system: provides a frame work / reference system for locating positions on a spheroid or ellipsoid. Positions are located using angles and angular units. "Earth model". When displayed in 2D, it's actually projected using something like the Plate Carree map projection.
Projected coordinate system: Two-dimensional framework / reference system for location positions. Includes a GCS as part of its definition. Has to distort data to crush the 3D surface into 2D: distance, shape, area using a map projection and parameters customized for a particular area.
NAD 83 / NAD 1983 / North American Datum 1983: a GCS built using data from Canada, US, ?and Mexico? Finished in 1986 before GPS really got going, there have re-adjustments since in both Canada and the US. Canada: CSRS. US: HPGN / HARN, NSRS2007, and 2011.
UTM: Universal Transverse Mercator. A grid system designed by the military. It's divided into 6 degree wide zones (there are a few exceptions) and covers the entire world. The map projection is transverse Mercator, with similar parameter values for each zone. The poles have separate UPS zones that use the stereographic projection. Can be used with any GCS, although the military uses WGS84 almost exclusively.
DMS is a string format, not a double value. You can store only one value per double precision field, but can certainly add two fields to a table! You may need to set the data frame to a geographic coordinate system first. Also check out the Convert Coordinate Notation tool.
Melita