Yes, those are good questions. For compatibility, there are not too many variables you have to worry about. I presume your old set has 40 or 42 tooth and 52 tooth chain rings, am I right? If so, if you get another chainset with a 52 tooth larger ring, then you won't have to move the front derailleur up or down, and you won't have to change the length of the chain. If you got one with a smaller front ring --48 teeth, for example-- you would have to move the front derailleur down, and you might want to shorten the chain a bit.

The square taper comes in two variants, and it is possible to get the wrong one. This mystifies me a bit, so I won't go into it.

Both chainsets and pedals come in either French or English threads. You probably have English (9/16").

If your chainset was exactly like the one in the photo above, then I'm pretty confident that Sugino Maxy chainset on ebay would be a perfect fit.

If yours has removable chain rings, like the one in the link you provided, then you also have to worry about the bolt circle diameter, BCD, unless your new chainset includes chain rings (as is common).