Learn Hardcore Polish: Nasz plan obejmuje spotkanie w muzeum i obiad w restauracji. - Our plan includes a meeting at the museum and lunch at the restaurant.

Polish has no articles like “a” or “the.” Definiteness comes from context. In w muzeum and w restauracji, you understand “in the museum” or “in the restaurant” just from the situation or previous mention.

What case is used after the preposition w in w muzeum and w restauracji, and why?

After w indicating location, you use the locative case (miejscownik).

muzeum is neuter and its locative form stays muzeum.
restauracja (feminine) becomes restauracji in the locative.
Locative answers “where?” (Where is the meeting? In the museum.)

Why do spotkanie and obiad look the same as their base forms? What case are they in?

They’re direct objects of the verb obejmuje, so grammatically they’re in the accusative case.

For the neuter noun spotkanie, nominative = accusative.
For the masculine inanimate noun obiad, nominative = accusative as well.
That’s why their forms don’t change, even though they function as objects.

What does the verb obejmuje mean, and what person/tense is it?