Could a stealth ship (like an Eclipse, Raven, or Sabre) bypass mines with relative ease? Will other ships be able to scan their surroundings to detect mines or is the Nautilus the only ship capable of doing that?

Any ship can detect (and be detected) by mines as it uses the game-wide signature and radar system. Stealth ships will naturally be harder to detect, so stand a better chance of safely bypassing mines. Ships with specialized radar/scanners will find themselves much better suited to detecting mines at distance.