"저는 노래를 합니다."
It's not. You misheard it.
Although, they are produced similarly in Korean, so it's easy to hear mishear it. In Korean, both ㄴ and ㄷ are produced with the tongue touching right behind the front teeth, so they can sound similar.
Also, the recordings they use on here come from some text to speech software and, well, they vary from passable to absolutely terrible.
Koreans don't do much voicing/humming before ㄴ as English speakers do before an n so ㄴ is often sounds like a hard n or soft d sound. If you pronounce 노래 as "Norae" or "Dorae" (with a soft d) people will understand you.
On a related note, this is also why ㄷ can sound like a t. Linguistically speaking, it's actually the same unaspirated t found in the word stop.
As a good rule of thumb, between vowels, ㄴ, ㄷ, ㅌ usually sound like the n in many, the d in edit and the aspirated t in top.
But in the first syllable of a word, usually:
ㄴ sounds like the n in no or a soft d. If you try pronouncing n and d at the same time you'll get the right sound.
ㄷ sounds like the t in stop (If you know Spanish or can distinguish Spanish t's from English t's, this is the same t found in the word "taco" when pronounced in Spanish).
ㅌ sounds like the t in top. Note that in English, this sound only ever appears when the first letter of the word is t.
To better distinguish the difference between ㄷ and ㅌ, try placing your hand over your mouth and saying the words stop and top several times. You will actually feel the difference and with training will learn to hear the difference as well. The hand over mouth trick is a nice trick to see if you're properly pronouncing ㄷ and ㅌ in a word.
Literally, no. But in reality, yes.
I'm singing should be "저는 노래를 하고 있습니다." or "저는 노래하고 있습니다."
But in reality, people use present tense as continously, so "저는 노래해요." often really means "I'm singing." just like "뭐 해요?" means "What are you doing?" and not "What do you do?"
"I can sing" is like saying ; I have the ability to sing but it doesn't say whether you are singing. "I can sing" would be translated to "부를 수 있어요". But to say "I sing" (like a hobby) it would be "저는 노래를 합니다" which is literally translated to "I sing do" or "I do singing". Hope that helped you!