Would someone please give an intuitive explanation of this? I can still visualise an end of the rod getting compressed and thus transmitting a longitudinal wave, but how can a transverse wave be created?
Strike the side of the rod, not the end. This leads to a transverse displacement wave.
Answer 1 is correct. But for a solid you have surface waves. A solid does not act like a fluid. Fluids only support compression waves. The solid has a crystal matrix that describes the reaction of the solid for all the different directions under influence of a force. In particular there is the Poisson effect which loosely speaking states that if you compress a solid in one direction it will expand in the orthogonal direction. Hence, if you strike a long rigid rod along its "longitudinal" axis you will excite waves in the orthogonal (transverse) direction. This phenomena is also described in seismology with S and P waves.