Translation:They say that one should never wipe tears without gloves.
"Torka aldrig tårar utan handskar" is the title of a triology by Jonas Gardell about the aids epidemic in Sweden during the 80:s.
Why does 'ska' mean 'should' rather than 'will' here? I guess it's because of the indefinite subject 'man', but is there more to say?
ska is a very versatile word, and occasionally, only contextual clues will dictate translation. This is probably one such case. There's simply no way a native Swedish speaker would mistake ska to mean "will" here.
American english language users prefer to use should but in G.B shall is used very much instead
What would be wrong in the meaning if we just put...the...before...tears...
I automatically said "gloves on", because the inference is that you would be wearing them.
That's fair - I'll add that. Please note that, strictly speaking, "gloves on" corresponds better to handskar på (sig).
There's a rule called BIFF which states that inte goes before the finite verb in subclauses. The rule's a bit broader than that, though - it applies to other disjuncts as well, for instance aldrig.