Learn Polish: Spacerował wszerz dziedzińca, bo szukał psa. - He was strolling across the courtyard because he was looking for the dog.

Wszerz specifically means "across the width of" a space and always requires the Genitive case (here, dziedzińca). Przez is more general, often meaning "through" or crossing over a boundary (like a street), and it takes the Accusative case. Using wszerz emphasizes walking across the physical span of the courtyard.

What is the base form of dziedzińca and why does its spelling change?

The base (Nominative) form is dziedziniec (courtyard). Because the preposition wszerz forces the Genitive case, the noun must decline. When it does, the mobile vowel -e- in the final syllable drops out, resulting in dziedzińca.

Is psa in the Accusative case here because it receives the action?

It looks exactly like the Accusative because pies (dog) is an animate masculine noun, but it is actually in the Genitive case. The verb szukać (to look for) is a strict exception that always requires its object to be in the Genitive.

Why are both verbs (spacerował and szukał) in the imperfective aspect?

Both verbs describe continuous, parallel actions happening in the past. He was in the process of strolling because he was in the middle of searching. Using a perfective verb would incorrectly imply a fully completed, one-time action rather than an ongoing state.