How do you feel about Shakespeare's use of the handkerchief? Is it a cheap effect just sprung upon us or does it grow from anything that has been said or done prior to its appearance?

The handkerchief is essentially a plot device. It doesn't actually turn up until Act 3 scene 3 but it serves as the focal point of deception, manipulation, paranoia and murder. I don't think it feels contrived in any way. The audience knows the evidence against Desdemona is pretty flimsy at best. This handkerchief just highlights this. It symbolizes much that is wrong with Othello; it illuminates his sense of ownership over Desdemona and his Paranoia over losing her.