Schwinn Varisty
 

I'm pretty sure this question is asked alot but here I go. I need a 3-piece crankset that would fit a 1-piece crank bottom bracket with the American-to-Euro adapter. One important thing is that I has to have a 42mm chainline and the cranks should be 165mm. So can anyone please help me find a 3-piece crank and 68mm x ___mm bottom bracket?

Well, here is an adapter that will let you fit a "modern" crankset at least. What year "Old Varsity" are we talking about, I assume it has the old American BB threads.
http://harriscyclery.net/itemdetails.cfm?ID=1335

1985 Schwinn Deluxe Varsity and yes, it has American BB.