No, a signal will not propagate along a single wire. It requires a transmission line with a fixed impedance, such as coax or twisted pair.
Incorrect. A single wire in free space _is_ a transmission line with a fixed impedance. The only reason we use coax and twisted pairs is to achieve a lower and controlled impedance, that does not vary depending on the proximity of random nearby objects. Also to reduce losses to free space.
You can easily demonstrate that a single wire works as a transmission line, using a fast pulser and good scope.
Edit to add: I think there is group delay in all kinds of transmission lines, induced by the characteristics of the materials involved. Skin effects in the wires, dielectric parameters of the medium around the wire(s), etc. Trying to minimize that is part of the art of transmission line design, no? I thought the only thing that doesn't exhibit group delay, is radio waves in a vacuum? (But it's not like I know what I'm talking about here.)