Learn Croatian: Ćelavi glumac je kupio šešir.

In Croatian, masculine adjectives have two forms: definite (ending in -i, meaning "the bald") and indefinite (ending in a consonant, meaning "a bald"). Since we are talking about a specific actor (the bald actor), we use the definite form ćelavi.

Why is the direct object šešir exactly the same as its dictionary form? Shouldn't it change in the accusative case?

For masculine nouns, the accusative case (used for direct objects) depends on whether the noun is living or non-living. Because šešir (hat) is an inanimate object, its accusative form is identical to its nominative (dictionary) form. If he had bought a living thing, the word would take an -a ending.

Where else in the sentence could the word je go?

je is an enclitic (a "weak" word without its own stress), so it strictly belongs in the second grammatical position of the sentence, right after the first noun phrase (Ćelavi glumac). You cannot start a sentence with it, but if you dropped the subject entirely, it would follow the verb: Kupio je šešir.