Yes. Adjective endings depend on the noun's gender and case. And they only decline when they come directly before a noun to describe it: - Ein rotes Buch. - Das Buch ist rot. (Not "Das Buch ist rotes.")
Also, the adjective takes a different ending depending on the type of article before it: - Das rote Buch. (Not "Das rotes Buch.")
No, that would be Hast du das rote Auto? I have a strategy for memorizing all this, if you're still looking http://gregreflects.blogspot.com/2015/02/how-to-memorize-german-cases.html
When the adjective is not in front of a noun, it doesn't need an ending.
When the adjective is in front of a noun, it needs an ending according to some rules.
Duolingo has an explanation for this lesson on the web version (scroll down): https://www.duolingo.com/skill/de/Colors
By the way, the English phrase is "doing my head in".
no determiner, accusative, neuter. m/n/f/pl: en/es/e/e https://www.lsa.umich.edu/german/hmr/Grammatik/Adjektive/Adjektivendungen.html