Which cartridge bottom bracked to replace cup and cone bottom bracket?

The modern Shimano cartridqe bbs are mostly symmetrical. 115mm would probably be best bet. 107 is asymmetrical as is one (or more) of the D- sizes.

Axles for cup and cone (and the cups) are still available in good quality. IRD recently had Tange produce a variety of these. There are other cheaper options too.

What a gorgeous bike that is!! I was a senior in high school that model year and that DX-3000 was my dream bike. Especially loved the black and yellow color scheme, which was the same as my state's university who's football program at the time was emerging from decades of losing into national prominence. I rode a Panasonic Sport 500 at the time, but made the move to college with a mountain bike in tow.

Much googling has not yet revealed a perfect cartridge bottom bracket for replacement for your crankset, an SR SXC-200 double, 52-42. You should measure the chain line--the distance from the center of the bike to the middle of the 2 chainrings. You could measure the distance from the edge of the B shell to the center of the chainrings and add 34mm (½ the bb shell width) or alternatively, the distance from the edge of the seat tube to the middle point between rings and add ½ the seat tube diameter (probably 27.2 overall diameter). The chain line should be confirmed to be 43.5mm

With this information, combined with your noted threaded shell width and spindle length, a bike shop with an experienced mechanic should have the correct cartridge BB in your hand right quick. Despite many hits on Google I'm simply not coming up with something viable. Shimano made a square taper bb, model BB-UN52, that apparently came with a 116mm spindle option, but I believe those are probably hard to find and certainly out of production. As you've noted, the axle's offset is yet another aspect tough to find.

I've compiled a few links to some helpful information for your bike. The Specs for 1986 Panasonic DX-3000 listed here are identical to ones posted in the catalog, which I've linked to below.

1986 Panasonic Bike catalog

Even the venerable, Sheldon Brown's Bottom Bracket size chart and primer is of limited help and no information is listed for the SR brands of crankset.

This link from First Components installation of square taper bottom brackets also has options for cartridge BB's that may be higher quality than the Shimano one discussed. Other manufacturers exist as well whose offerings are both above and below Shimano level quality. One note: while this is an excellent tutorial on the installation of cartridge bearing bottom brackets with a brief primer on square taper spindle interfaces, it has one problem. The aspect of the tutorial I vehemently disagree with is the suggestion to utilize a torque wrench to break free the threads of the old cups from the frame. That is monumentally stupid advice even with the torque setting on high. The rationale given is the increased power afforded by the long handle of the torque wrench. A breaker bar or cheater bar slipped over a different wrench should be used for increased leverage. NEVER use a torque wrench in that manner.

With the info you've collected and confirmation of the 43.5mm chain line, you certainly should have what's required for a shop to set you up with a workable cartridge BB.