Do all planets have a molten core?

Depends on what you call "planet." For example, in comparison to Earth, the Earth and Moon is a "binary planet system." And the moon doesn't have any molten core. Also, you said inner. Inner core of Earth (and many other planets) are solid because of (what I think is pressure). Finally, if it is outer core you meant, and you didn't count the moon as a "planet", would you count extrasolar planets as "planets". Please clarify, but to your current question, no.