Learn German: Das traurige Gespenst soll im Wald weinen. - The sad ghost is said to cry in the forest.

This is the subjective use of sollen, which you use to report a rumor or hearsay. When someone says Das Gespenst soll weinen, they aren't giving the ghost a command ("The ghost should cry!"). Instead, they are distancing themselves from the statement and reporting what other people claim: "People say the ghost cries" or "The ghost is said to cry."

Gespenst is a neuter noun (das). Shouldn't the adjective be trauriges?

Because we already have the definite article das right before it, the neuter gender of the noun is already clearly marked. Therefore, the adjective takes the "weak" ending -e in the nominative case, making it das traurige Gespenst. You would only use the strong ending -es if the article didn't show the gender, such as with the indefinite article ein (ein trauriges Gespenst).

Why do we use the dative im Wald (in the forest) instead of the accusative in den Wald? The ghost is doing an action (crying), right?

It's true that crying is an action, but with two-way prepositions like in, you only use the accusative case when there is movement towards a new destination (answering the question "Where to?"). Here, the crying is happening entirely within a fixed location (answering "Where?"). Because there is no change of location, we use the dative case (in dem merges into im).

I usually see Geist for "ghost". What is the difference between Geist and ?