Dating a Schwinn Varsity is easy. The first letter of the VIN # corresponds to the month of build (A=JAN, B=FEB, etc). The second letter is the year (beginning with '65 as the letter A, '66=B, etc). However, "I" and "O" were not used, as they could be confused with the numbers "1" or ""0".

This link is to vintage Schwinn catalogs and dating/production number help.

As for frame sizing, Schwinn used a dimension from the center of the BB to the top of the seattube. This dimension is irrelevant, when it comes to proper fit. The effective toptube dimension is more important. On a 24" Varsity, the ETT is 59cm. On a 26" Varsity, the ETT is 62cm. I have one of each size. If yours is a 26", it's probably pretty close on proper size. There are some options to gain more cockpit room.

As for your handlebar dilemma, compatibility is an issue, anytime you're trying to add new parts to the mix. The stem adapters here are for 22.2mm ID fork tubes (Varsity is 21.15mm). I haven't tried it, yet, but one could machine the fork ID to the 22.2mm necessary to fit an adapter. The Varsity steertube is incredibly thick, compared to other bikes. I wouldn't be afraid of doing that, as my English bikes are 1" steertube with a 22.2mm ID. Using such an adapter would allow one to use a plethora of modern stems, in multiple lengths/angles, with a 26.0mm clamp. Original is 93mm length and -14deg angle. Browse the catalog I linked above.

Also, Truvativ makes an adapter to allow one to use a modern, sealed-bearing BB and crankset. Wheels Manufacturing makes a shim to allow one to use a modern, front derailluer. The rear derailluer hanger will accept modern, rear derailluers, as it's metric threaded. Tektro offers model #556 long-reach, dual-pivot calipers that'll work. Modern, 700c rims slide right in. Wald offers a seatpost that is flared to 7/8", at the top, to allow one to use a modern, seat clamp. Modern, sealed, traditional-fit, 1"-threaded headsets are available. Only your imagination and wallet limit the scope of your project.

I hope that helps with your endeavor.