Yes, the spokes are always in tension unless something catastrophic happens.
Yes, string would work (and they do, if you look at Spinnergy wheels with Kevlar spokes). The strings do have to have good tensile strength though, otherwise the scenario becomes silly.
I don't like the 4-spoke model because it is not realistic. You will have to now define the stiffness of the rim, whether or not the spokes can take load in deflection and whether or not the spokes are rigidly pinned to the hub and rim. Don't go there, it'll not help your understanding.
If it helps, hold a wheel in your hands by holding the axle vertically. Now push on the tyre, towards the hub, with your other hand. Note between which two points the forces act. Note how no other point on the wheel is involved. It is easier to visualise the effect like this than on a vertical wheel.