"I do not want this chicken."
Translation:Я не хочу эту курицу.
No it can't be. After the verb хотеть fellows the accusative case. And the accusative case changes the ending of female nouns from -a to -y. For more info see http://www.russianlessons.net/grammar/nouns_accusative.php
I can't think of a definite rule when you should drop the pronouns and when not. It depends on the context. Here, the basic translation is with the pronoun, you'd better remember it as the default.
Unlike Spanish, it is not common to drop almost all personal pronouns in Russian.