Forewords: As Heidar remarked in the associated comments, my answers were not dedicated to the topological insulator situation. I'll try to correct myself in some edits I'll write [-> into brackets <-] and into answer-bis, but I let my answers about topological superconductors, since they may be helpful.
Short answers, please separate your questions if you want more detailed answered:
1) Dirac cone's on surface: Some emergent Dirac cones appears in the bulk of the $p$-wave chiral superconductor, see the book by Volovik for more details, available freely on his homepage at Aalto University. I'm not at ease with the notion of band structure on the surface. I have no idea what it means... That's just the closure of the gap which happens on the surface/edge for me. [-> Please see the Heidar comments for a clever discussion <-].
1-bis: the topological insulator situation. The topological insulator case is easier to discuss, since a bulk insulator has no closure of the gap by definition. Then, the Dirac-linear-closure can only happens at the edge. See also point 4 below, and the Heidar's comments about the Jackiw-Rebbi model below.
2) Shape of the cone: The shape, per se is not important. What you need is a linear dispersion relation with a crossing point. (NB: Without crossing, the dispersion corresponds to the Weyl fermion particles.) The cone structure is the simplest structure like this.
3) Symmetry protected topology: I don't know the full answer to this question. I would say no, not for the emergent Dirac cones in superconducting/superfluid phase: the cone can there be topologically protected as well. But the topology depends strongly on the symmetry for quadratic Hamiltonians, especially the three discrete ones of particle-hole $P$ such that $\left\{ P,H\right\} =0$ with $P^{2}=\pm1$, time-reversal $T$ such that $\left[T,H\right]=0$ with $T^{2}=\pm1$ (both $P$ and $T$ have anti-unitary representation, and $H$ is a representation of the Hamiltonian), and the chiral $C\equiv PT$ ones (a situation exists when $C$ is present without neither $P$ nor $T$). This is still troubling for me. I think it's essentially a matter of convention whether you want to call these discrete symmetries some kind of topology (whatever it means) or not. Topology for me means you've got a Chern number $\nu\neq0$, and you will keep it until you change one of the discrete symmetries I mentioned. But some Chern numbers are protected by symmetry as well, so it is a mess to disentangle all these notions at the end.
3-bis: the topological insulator situation. For the topological insulator once again, the situation is easier, since the topological classification is crystal clear: the topological characteristic are provided by symmetry. These symmetries are just the three discrete symmetries I discussed in point 3.
4) Opening <--> closure of the gap I think the answer to this question has been answered long ago by Volkov, and Pankratov, Two-dimensional massless electrons in an inverted contact JETP, 42 178 (1985) (article for free) or I misunderstood it. The answer is yes, and you get an instanton solution at the boundary, as in the Jackiw-Rebbi. Volkov and Pankratov discuss the Dirac dispersion relation, not a relativistic model.
