Many different ways to use your thumb, but not resting it on a string is about letting the forearm rest against the bass rather then on it, or leaning on it. The floating thumb technique has a great sticky on the techniques section devoted to it.
String thickness is not such an issue as string tension, modern strings are wound around a core so the thickness is not always related to tension ( the string being stiff to support it's own weight...the core does that ) Since it is the underside of the string that makes contact all strings should have roughly the same gap to the fretboard, so no single string is any different to fret.....the underside of the strings roughly have the same gap to the fretboard.
Obviously you have to find a string that suits you, the same is with your thumb.
Check out the link