The choice of datum doesn't matter for map making (provided you change datums appropriately, of course). It does matter for transmitting coordinates and sharing data.
To address your last question ("does it even make sense?"), note that UTM is really a coordinate system and as such--in addition to its grid-based zone naming system--it includes a definite datum and unit of measurement as well as a set of projections. What we're talking about here is using its projections (Transverse Mercator (TM) for the low to moderate latitude zones) and its zoning system but possibly changing the datum (and, perhaps, units of measurement). Changing either the datum or the units (or both) is fine, but after that is done the result is not "UTM," strictly speaking. In some applications (e.g., military targeting) it could be dangerously misleading to refer to the resulting (x,y) values as "UTM coordinates," because they will look remarkably like UTM coordinates but might be up to a few hundred meters off. (That can lead to embarrassing things like lost embassies.) Nevertheless, because your principal questions refer to "UTM" in this looser sense, I will also use the term in this way.
UTM is a choice: you can pick any zone to use. It is designed to work within certain accuracy constraints to within 3.5 degrees (longitude) of its meridian at any latitude within the 120 non-polar zones. Because the zones are nominally 6 degrees wide, this allows a 0.5+0.5 = 1 degree longitude overlap with each neighboring zone. Moreover (if you are concerned with map quality foremost and following the UTM system exactly is of lesser importance), the overlap can get larger in the more extreme latitudes (e.g., at 45 degrees the overlap is more than 4 degrees of longitude). Because two (reasonable) datums differ in longitude by at most by 100 meters or so (0.001 degrees), the difference in datum is inconsequential: if you choose a UTM zone based on one datum it will work fine with another.
Once you have allowed yourself the flexibility of changing the datum, you might as well go all the way and create a "custom UTM zone." This is tantamount to dropping UTM altogether (which includes forgetting about the 0.9996 scale factor and origin offsets) and selecting a TM projection suited for the area you are mapping. Within a small area you can do well by placing the meridian of the TM projection through the center of your map and using a scale factor of 1.0000. Choose any convenient origin.
I haven't looked at your code, but in the URL you provided, your X and Y are flipped around. For Faroe Islands, X should be -6.7866 (longitude) while Y should be 62.01387 (latitude). Here's a working URL of what you're trying to do:
http://sampleserver3.arcgisonline.com/ArcGIS/rest/services/Geometry/GeometryServer/project?f=json&outSR=32629&inSR=4326&geometries={"geometryType":"esriGeometryPoint","geometries":[{"x":-6.7866,"y":62.01387,"spatialReference":{"wkid":4326}}]}&callback=dojo.io.script.jsonp_dojoIoScript2._jsonpCallback
That correctly returns:
dojo.io.script.jsonp_dojoIoScript2._jsonpCallback({"geometryType":"esriGeometryPoint","geometries":[{"x":615863.053581134,"y":6877701.89119242}]});
As for 500s, I seem to get those quite frequently with sampleserver3 as well tonight. Try using sampleserver1 instead, that one seems to be more stable.
Best Answer
The UTM zones and their respective EPSG codes cover the whole area of their respecive zone from the equator to 84N / 80S. These areas are subdivided by the letters from South to North, and these again are subdivided into 100km grid squares, but the projection system is not affected by these subdivisions. Hence for 29U you can use EPSG:32629.
For areas above 84N / below 80S you should use the Universal Polar Stereographic coordinate system.