Is the string-net model Hermitian?

The string-net Hamiltonian $H=sum_{p}(1-B_p)+sum_{v}(1-Q_v)$ is Hermitian, this is because that $B_p$ and $Q_v$ are all projectors. The fact that $Q_v$ is projector is obvious from the definition of $Q_v$. The argument for $B_p=sum_kfrac{d_k}{D^2}B_p^k$, where the sum runs over all labels of simple objects in unitary fusion category $mathcal{C}$ (denoted by $k$, $d_k$ is the quantum dimension of $k$, and $D^2=sum_{k}d_k^2$) and each $B_p^k$ inserts one $k$-loop into the plaquette $p$, is more technical. But it can still be checked that $B_p$ is indeed a projector by carefully do the graphical calculus on one plaquette $p$. Since adding two concentric loops can be calculated by $B_p^kB_p^l=sum_{q}delta_{q^*kl}B_{q}$, then it is straightforward to check that $B_p$ is indeed a projector. There is no need to use extra mathematical results but direct calculations. The inner product of morphism space (and trivalent vertices) is defined in this way such that the morphism vector is normalized and $F$-matrices are unitary.

We should recall that the Hamiltonian is a construction that satisfies hermiticity. The Hamiltonian is not a derived mathematical object from some theorem.

If you follow Kitaev and Kong's exposition, then I think you are right. The hermiticity of the Hamiltonian results from the hermitian inner product of morphisms defined in eq.(11). If you follow the physical motivation of the "fix-point" Levin & Wen model(s), then hermiticity is enforced because we want the Hamiltonian to be hermitian, which in term defines a hermitian inner product on morphisms (this coincides with the defintion in Kitaev and Kong's paper).