There are two issues: how someone speaks and what you hear. There may be several different ways of transcription and it is not always easy to find a consensus.
Here you have some samples of spoken Polish:
Because both are singular. The words change depending on their function in the sentence. "psa" and "pies" are different cases.
"pies" is Nominative - the basic, dictionary form. It's mostly used for the subject of the sentence.
"psa" is Accusative (and Genitive). Accusative is used for the direct object of the sentence. Almost all verbs that you learn at the beginning of this course need Accusative.
I know this is a Polish course, not an English one but the "correct" answer – dog does not write – is not correct English. The sentence has to begin with an article. However, would the phrase in Polish mean that dogs are not capable of writing ? In which case, "dogs do not write" should be a good translation