A metal core is advisable because it concentrates the magnetic field. This makes magnetic coupling between coils more effective, but you can do it without the core too, especially if both coils are tightly coupled (mechanically close to each other). If you don't have a core and the two coils are some distance apart, part of the generated field will pass outside of the other coil.
(In a transformer the outside metal "shortcuts" the field, increasing it. Without this outside metal the field would fill a large volume around the core.)
The static magnetic field itself will not induce any current or voltage in a wire. (Else we would have been able to use simple wires as magnetometers for our smartphones, in stead of the complicated chips they have now).
The only effect you will notice is, as you say the EM wave(s) created by the cable coupling into anything they can couple into with the right directionality and causing some loss. This, however, is even negligible if you run a wire above a copper plate for the more likely frequencies. If you go into radio frequency and such those effects might be noticeable on their own, but then you're already also in the domain of skinning and parasitic capacitance and all such, which I would expect to be more pronounced anyway.
Now, if you were to take away the skinning effect and make a few loops, let's say 1.5 or more, you can couple into some metals, but then we'd just call that an inductor to begin with.
I have not, to date, run into a situation where I had to think about static magnetics near a normal straight wire. But that's possibly because any magnetics placed anywhere in my designs usually interact with a coil quite near them on purpose.
But as I am too tired to do any maths to prove it, I'm even more open to standing corrected than usual.
Until further notice, I'd say, just go with resistivity, unless you are working with HF, in which case you are going to have much more serious problems with other things that a permanent magnet here or there.
Note: At 10kHz something in the AWG15 or 16 range will already show a noticeable skin effect, so even outside of radio frequency, if you go beyond 2 or 3kHz at higher power levels, it can be very wise to look up the term "skin effect" on Google.
A Rogowski coil is not a current transformer.
The current in the main conductor generates a magnetic field round itself. The Rogowski coil samples the field, generating a voltage as the field changes. It is weakly coupled to the main conductor. As little, ideally no, current flows in the Rogowski coil, there is no effect back on the main conductor.
Any loops couple. The sampling coil would generate a voltage whether it formed a closed loop around the main conductor or not. However, that's not terribly useful, as the coupling, and hence the generated voltage, would change if the conductors changed position. The sampling coil would also respond to any current changes anywhere in the vicinity.
The reason the Rogowski coil takes the form of a uniform toroidal winding is because due to the symmetric situation, it couples to all the current flowing within the toroid, regardless of the position of the conductor. There is zero coupling to any current that does not pass through the hole. An incomplete circle, or a non-uniform winding, would not have these two desirable properties.