Remember that Russian does not have articles (a/the). этот/эта/это/эти are demonstrative adjectives that are used in Russian often to the same effect as using the English definite article "the" to draw specificity to otherwise indistinguishably different objects. As adjectives, they are set in grammatical agreement with the nouns they modify in case, gender, and number (masc, fem, neut, plr respectively).
Many learners of Russian are thrown off by the appearance of это early on as a pronoun in a "this/that is ___" construction. Это is both a demonstrative adjective and a demonstrative pronoun. Remember that the verb "to be" does not exist in the present tense for most declarative sentences. That's why you see Это стол., Это кошка., Это окно., Это дома. Although each of these sentences contain only two words, they are indeed complete sentences. Note that each of them has a different gender and the final one has a plural noun to show the contrast from the adjective это. They are complete sentences because they have a subject and a predicate. Nouns can certainly be in the nominative case in the predicate. It's called a predicate nominative, and it also happens in English. It works the same way as saying Он врач, "he is a doctor". Both nouns are nominative, but the subject of the sentence is "he" and the predicate is "doctor".
You have the right idea that the sentence is specifying a particular exercise by saying "the", but этот/эта/это/эти always equate to this/that/these.
In the English for Russian speakers course they use "этот" and its forms to help the Russian natives learn to use direct articles. I take it that this is mostly a sort of kludgy pedagogical crutch, but I'm not sure if there might be any useful knowledge for learners of Russian (from languages with articles) to be extracted from it.
Both are fine ? Obviously this sentence, inasmuch as it contains "this," can't exist (outside Duolingo) without a context. That context may well demand either "do" or "am doing," but a priori, it's impossible to know which is more likely. I would note in passing that Ngrams will give you a nice graph for "I do this exercise," but has nothing for "I am doing this exercise."
"I do this exercise" just sounds wrong to me in English. "I do this exercise" I guess would work if you do it regularly, but that seems like an unusual situation and would require more info, "я делаю это упражения кажды ден". If you are currently doing it, then it would have to be "I am doing..." if you will do it in the near future, wouldn't that be "я буду делать..."
Right, "this / that" and their translations are called "deictic expressions," meaning they always demand a context to have meaning.
I think "I am doing this exercise" sounds a touch odd because for it to have meaning, someone would have to be be seeing what the "this" is, but then why, really, would you have to tell them you're doing it if they can already see you doing it?
Saying "I do this exercise" actually imparts useful information: that you do it on some sort of recurrent basis, a fact which the situation itself does not make clear.
Now, if you're going up to your teacher to say "I am doing this exercise [either then pointing to it, or referencing the упражнение that everyone else is also doing], and I need help understanding blah-blah-blah," then obviously it's not "I do," but that's a rich context, too.
So I'm actually not right about "I am doing" being odd. It's just that the context that jumped into my head is a guy doing deep knee bends to show off his exercise regimen. The schoolwork example is no more or no less plausible. Who would have the slightest idea which is more common? (although NGrams happens to imply it's the "I do..." version)
Am I the only person for whom the audio seems a bit off? I mean - isn't the person saying делал rather than делаю?
Definitely doesn't seem as clear as it could be. I hear it as "делаю," but there does seem to be some distortion of the "а" sound.