Да! Accusative, since it's the direct object; neuters don't change in the accusative. Careful: animacy in the accusative only matters for masculine singular, and all plurals. Feminine singulars in -а/-я do change, whether animate or not: Я люблю мою подругу, and Я люблю эту книгу.
Russian word order is really flexible, so that's just something to get used to. It's not random, though; unlikely German, where you can make a real mistake in word order, in Russian it has more to do with context and tone.
To generalize: in a neutral tone, new or emphasized information tends to come last; in a more expressive or conversational tone, that first tendency is flipped so new/emphasized information is not last (that's how I interpreted Я много раз это говорила).
A few videos on word order are at:
Another good explanation is here: http://www.study-languages-online.com/russian-words-order.html
To expand on their examples at the end, some context can help:
Куда едет Мария? -Мария едет в Москву. Where is Maria going? -Maria's going to Moscow.
Кто едет в Москву? -В Москву едет Мария. Who is going to Moscow? -Maria is going to Moscow. [or: It's Maria who's going.. or: Maria is the one who's going...]
It seems a little bizarre to English speakers but the only significant change requires placing the pronoun (in this sentence, 'this') before the verb instead of after.
In English, it is common enough to put an adverb after the subject and before the verb. For example:
I often say this.
In Russian, it seems most adverbials commonly do this, where in English it wouldn't sound natural. So just imagine Russian adverbials work like 'often' in English.
NB. I'm only a learner though.
Это is nominative (or accusative, as in this example). Этого is genitive (masculine or neuter). Genitive has a number of uses, including possession/ownership (паспорт этого студента = this student's passport), after certain prepositions like у (у этого студента = at this student's place, or 'this student has...'), and absence/nonexistence (У них этого учебника нет = they don't have this textbook); sometimes also for a negated direct object (я этого не видел = I didn't see that).
It is a little disconcerting to hear this sentence (and several other recent examples with a female subject indicated by a female ending on the verb) delivered by a deep obviously male voice, which has just appeared as second available voice in the last few weeks. Does the Duolingo basic framework give the possibility to specify a male or female voice for the relatively few sentences where the gender of the speaker is specified by the content?