Can someone explain why it is эту instead of perhaps это Does it have to agree with the accusative?
Well, first, in nominative case it would be эта курица, not это. But, yes, эта becomes эту in accusative case. Basically all of the forms of demonstrative pronouns have the same declension as nouns of corresponding gender: этот, эти stay the same in accusative as in nominative unless the noun is animate, in which case they follow the genitive (become этого, этих, respectively), just like for nouns.
So how do I know if to use genitive or accusative in a case of negation with transitive verb?
In another exercise, it was stated that objects of negated transitive verbs are cast in accusative rather than genitive if they are living organisms. Thus:
Я вижу книгу - I see the book
Я не вижу книги - I do not see the book
Я вижу курицу - I see the hen
Я не вижу курицу - I not see the hen
You determine the case/gender/number of the noun and then use the appropriate determiner. The ends of determiners do not have the same endings as the nouns:
Эта муха - this fly
Эта лошадь - this horse
эта деревня - this village
Here's a very useful set of declension tables for pronouns and determiners. I highly recommend this particular page, as it will provide a lot of resources in figuring out pronouns, possessives, determiners, demonstratives, interrogatives.
Up until now курица was translated as "hen". This is my first instance that "hen" is taken as wrong.
That's because in Russian - we have word "курица" - as for a bird and for food. Translated it into English - when you're speak about food - it would be chicken, and if you mean bird - it translated as hen.
I believe it would be inanimate, but as this distinction is only important for masculine nouns, it doesn't really matter.
Here, «курицу» is in accusative case due to the transitivity of «хотеть», right? Is genitive case («курицы») only forced in possesive negation?
I've seen cases where you don't need the "Я" to say "I don't want..." I've seen people just say "Не хочу" does this work for all cases?