Learn Polish: Ten znany piekarz powinien wygłosić przemówienie. - This famous baker should deliver a speech.

In Polish, you don't 'make' (zrobić) or 'give' (dać) a speech. The required collocation for delivering a speech, lecture, or presentation is wygłosić (perfective) or wygłaszać (imperfective), which translates closer to 'to voice out.'

Could we use the imperfective verb wygłaszać instead of wygłosić?

You would only use the imperfective wygłaszać if the baker was supposed to deliver speeches regularly as a habit, or if you were emphasizing the continuous process of speaking. For a single, completed event, the perfective wygłosić is needed.

What case is the noun przemówienie taking here, and why does it look unchanged?

It is in the Accusative case because it is the direct object of wygłosić. It looks identical to its dictionary (Nominative) form because all neuter nouns ending in -e, -ę, or -o share the same form for both Nominative and Accusative.

How would I add the phrase 'on behalf of the bakery' to practice this lesson's compound preposition?

You would use the phrase w imieniu followed by the Genitive case. The word for bakery is piekarnia, and its Genitive form is piekarni. You could add w imieniu piekarni to the end of the sentence to say he is speaking on behalf of the bakery.