Q: Is every compound Poisson distribution a mixed Poisson distribution?
A: No. Every mixed Poisson distribution has a variance greater than or equal to the mean. The compound Poisson distribution is not so constrained.
If the i.i.d. random variables that are compounded have mean $\mu$ and variance $\sigma^2$, then the compound Poisson distribution has mean $\lambda t\mu$ and variance $\lambda t\sigma^2$ (adding $N(t)$ variables with rate $\lambda$). So the variance will be smaller than the mean if $\sigma^2<\mu$