Is frequency quantized in the black body spectrum?

Frequency is not quantized, and has a continuous spectrum. As such, a photon can have any energy, as $E=\hbar\omega$. However, quantum mechanically, if a particle is restricted by a potential, i.e.

$$\hat{H}=-\frac{\hbar^2}{2m}\nabla^2 + \hat{V}$$

for $V\neq 0$, the energy spectrum is discrete. For example, in the case of the harmonic oscillator,

$$E_n=\hbar \omega \left( n+\frac{1}{2}\right)\quad n=0,1,2,\dots$$

"Yes", but the quantisation depends on the size of the box. In practice the 'box' is large and of variable shape, so all sizes are available, so all frequencies are available.

Ultimately, it is somewhat of a philosophical question who's answer depends on which axioms and base concepts you (they) are using at the various stages of reasoning. Consider, does time pass for a non-interacting particle? Can a non-interacting particle be kept in a box? etc.

Try for a quirky view on the problem. Link Between the P≠NP Problem and the Quantum Nature of Universe