"I do not sing songs."
"うたいません。" is the most natural answer. I suppose you could say "I don't sing songs" doesn't necessarily mean "I don't sing" since you could sing gibberish that you make up and claim it's not a song. It's implied that you don't sing songs, so you don't have to say "I don't sing songs." At any rate, if you wanted to translate into something that's not quite native level you'd say "歌をうたいません。" Unless their teaching point is that the verb, to sing, in Japanese is intransitive; that would make the sentence much more difficult to translate. "を" is the object marker and songs is the object of this sentence.
For native Japanese, just "歌いません" is a natural answer because if I listen "sing songs", it translates to "歌を歌う", so it could be separated as "歌を" and "歌う", in Japanese sometimes nouns are omitted. So if you are listened "歌いません", it means "do not sing songs" then it means "歌いません".
See here for most of them: https://www.freejapaneselessons.com/lesson04.cfm
は is a topic marker. It marks the theme/topic of the sentence.
が (not か) is a subject marker. It generally marks the noun that is performing the action/verb.
を is an object marker. It marks the direct object of the sentence - the noun being acted upon.
に has a bunch of different uses; it can be an indirect object marker (e.g. the person you're writing a letter to), a direction marker, or a time marker.
で also has several uses; it can be a location marker (i.e. where an action/event occurs), or it can describe a method/means (i.e. the means by which an action is done or takes place). There are a few other uses (e.g. marking the material something is made out of, or indicating a sum), but these two are probably the most important to know.
The casual form of the verb is 歌う.
To make it polite, we change the う to い, and then add ます
歌う → 歌います
This is just part of the verb, different from the います in previous exercises
The same happens for many other verbs too
Drink: 飲む → 飲みます (mu becomes mi)
Read: 読む → 読みます (mu becomes mi)
Write: 書く → 書きます (ku becomes ki)
But that's only, "I don't sing" however the question was "I don't sing songs" so you should add songs in the translation. "I don't sing songs" is not necessarily the same as "I don't sing." My son sings all the time, especially when he's eating, but rarely does he actually sing a song. My crystal sings, but it doesn't sing a song.
It is my understanding that adding い turns something into an attribute, like あお is the color blue and あおい is the attribute of being blue. So as far as I understand it the Japanese literal translation for this exercise is not "I don't (perform the act of) singing songs", but rather "I don't (have the attribute of) singing songs". Do I understand this correctly ?