It will sound nicer with snare cord rather than plastic straps. I went back to cord a year or so ago and won't go back to anything else. The cord lasts too. No wear at all on the cord I put on the snares back then--Ludwig's Gold-ish colored cord.
Also, check to see that the 2 heads are working together to create a NICE ring, and not some obnoxious "poing" that goes up, or dips down after you hit it.
Ludwig heads can sound good, but it seems to be easier to get a sound you really like faster with other brands (Remo, Evans, Aquarian). I used Ludwig's heads for a while, and they did sound nice, but it WAS different sound than I was used to, and took some fiddling to get what I wanted. They didn't wear out very fast though, and the coating is good stuff.
The tone of the 402 shell can be a bit long, so don't be afraid of a little "extra" compared to some other snares.
Sliver of Moongel will tame it down a bit if you need that sort of sound, but you don't need a whole slab, just a slice of it, maybe an 1/8th of an inch wide toward the edge (like around the brand logo). Any more than a small piece of that stuff will just kill any drum IMO.
Ludwig's wires are fine, but other wires, such as Puresound Customs do add to get a little more "solid" a sound. I personally like some "dirt" in my snare sound, so I don't mind cheaper wires sometimes
As long as it's not "over the top ring" I hear through the monitors, or the mains I'll leave the snare open. I might move the mic a little further off the head to hear that first, but I'm not opposed to a sliver of Moongel.
Have fun experimenting!