I found the following slides helpful: https://physics.princeton.edu//~haldane/talks/dirac.pdf
Here are my answers:
Yes. From what I understood, the above is not necessarily a 3D effect. The wormholes appear only at gapless points, so we can expect only one Dirac string in my example.
Partly correct. We have only one wormhole from the gapless point. However, ...
I don't think I can use this Hamiltonian to recover the band structure. The band structure should use a tight-binding Hamiltonian with several unit cells. A colleague told me that this Hamiltonian is just for their discussion in Figure 4.