Breadboards are a tool to allow for rapid prototyping without semi-permanent applications like crimping or soldering. They are not required.
Jumper wire, which is typically 20-24 AWG wire, is ideal for small current applications like a single led.
Depending on how you mount that led, you should consider heat shrink or any other type of non-conductive covering. Tape works. Hot Glue works fine. If you mount the led in a panel or hole of some type, so it doesn't flop around, then you really don't need to insulate the leads, but it still helps.
Heat shrink or other insulators do help improve the mechanical strength of bare wire. Solid wire is stronger, but more brittle than stranded wire. Even with heat shrink on, solid wire will break at the bend point if you bend it too much (metal fatigue), while stranded wire will last longer (but will break eventually too). Heat Shrink makes both last longer.