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 first moment $\mu$ and second moment $\tau^2\geq\mu^2$, then the compound Poisson distribution has mean $\lambda t\mu$ and variance $\lambda t\tau^2$ (adding $N(t)$ variables with rate $\lambda$). So the variance will be smaller than the mean if $\tau^2<\mu$.