yep, if you're not worried about keeping the bike 100% period correct, then you're best off getting a sealed BB. You say you have 600AX cranks - are you sure they're that model (FC-6300) and not 600EX (FC-6200/6206/6207/6400)? The AX version has extra large holes for the pedal spindles, and require dedicated pedals, which if you don't have already might be very hard to find.
Look on the back of the crank arms and they should have the model number stamped somewhere, for example "FC-6300" (for 600AX). once you've found the model number, it's easy to know what BB length you require.
you said you're BB is marked 1.37x24, so in that case it's english threaded. From velobase, 600AX cranks require a 116mm spindle if used with an english BB. So any english threaded BB with a 116mm spindle should work fine. if you can't find 116mm, then you'll be fine 1 or even 2mm difference either way, but if you go shorter then make sure you check the clearance to the chainstay! if you get a shimano BB then no problem at all, but the majority of BB units are for shimano (JIS) taper as opposed to campagnolo (ISO).
you only need a single tool (and large spanner!) for most shimano style sealed BBs.
and yes - keep the old bits just in case! personally I never throw anything out unless it's absolutely b*****ed, you just never know when you (or someone else) might need them!