plot legends without border and with white background

Use option bty = "n" in legend to remove the box around the legend. For example:

legend(1, 5,
       "This legend text should not be disturbed by the dotted grey lines,\nbut the plotted dots should still be visible",
       bty = "n")

As documented in ?legend you do this like so:

plot(1:10,type = "n")
abline(v=seq(1,10,1), col='grey', lty='dotted')
legend(1, 5, "This legend text should not be disturbed by the dotted grey lines,\nbut the plotted dots should still be visible",box.lwd = 0,box.col = "white",bg = "white")
points(1:10,1:10)

Line breaks are achieved with the new line character \n. Making the points still visible is done simply by changing the order of plotting. Remember that plotting in R is like drawing on a piece of paper: each thing you plot will be placed on top of whatever's currently there.

Note that the legend text is cut off because I made the plot dimensions smaller (windows.options does not exist on all R platforms).