The sheet music on Masato's link gives the first verse as:

Should auld acquaintance be forgot
And never brought to mind?
Should auld acquaintance be forgot,
And auld lang syne?

which should settle the question of "for sake of" versus "and days of" auld lang syne. But that isn't going to stop me from discussing it!

"For the sake of auld lang syne" makes no since at all IMHO; why would you forget your old friends for old times' sake, unless they've recently turned into axe murderers and you want to keep the positive memories of long ago? It would make more sense to say, "Shouldn't old acquaintance be remembered, for old times' sake?" which doesn't quite scan :-).

I have always heard and sung the last line (a US thing?) as "And days of auld lang syne," which does mean something: Should old friends be forgotten, and should the days of long ago be forgotten? (the days long gone, days that are long since gone -- you can't quite translate it literally can you?)

Well it looks as if neither is "right" but I don't guess I'll change the way I sing it.

Oh dear. Not on topic, but up above I originally wrote, "It would make more since to say, ..." Oh dear. Age is creeping up. Once the brain cells go it's the beginning of the end.... Well I hope my friends will be patient with me, for old times' sake!

Rita