Plural would be «Ми шука́ємо соро́чки». For inanimate nouns, plural accusative is always the same as plural nominative. For animate, plural accusative is always the same as plural genitive.
I'm OK with «ти». :) In some environments, «ти» is the default. For example in the universities many students use «ти» when speaking to other students, even if they don't know each other, (but not to professors) — and Duolingo can be a compared to a virtual university. :)
But you should have used «твою́ ві́дповідь» or «Ва́шу ві́дповідь», because «ві́дповідь» is feminine.
No, they're not vocative, they're accusative. The pronoun «за» is used with accusative case when it means 'for', so «твою́/ва́шу відпові́дь» is accusative.
You could use vocative case for «мій друг», because this is a form of address. Vocative would be «мій дру́же». However, nominative works here too.