"My car is big."
Adjectives in Japanese are somewhat more complex than in English, and fall into two broad categories. The first I call "な" adjectivesas they work like possessives and attach to the front of nouns with な, or occasionally, の. 青の皿 "Blue's plate" is a blue plate for example. The other class is more subtle, but ALWAYS ends in an い sound. They work as verb-like adjectives. They can conjugate into all the verb forms, and may end sentences. If they do, they are assumed to modify the topic/subject of the sentence, and no other true verb is needed. However, in polite situations, a purely ornamental です may be attached. There's a lot more to this topic, and I doubt I got all the subtleties right, bit that's why this sentence has no です
In this specific case the subject of the sentence is specifically you. You cannot simply attribute an adjective to an object without subject in japanese, otherwise you're talking broadly 車は早い "cars are fast", this is also the case for all other cases where adjectives are shown, and there are other example sentences on duolingo.
This is the general rule of thumb, and it's not always correct, but works most of the time. If it's just kanji, more often than not it's on'yomi, and if it's mixed, it's often kun'yomi. Sometimes that doesn't help you very much when kanji has 5 separate pronounciations, like 女 for instance.
Ownership is inferred by possessive の whereas は highlights the topic here: 私の車は - speaking of my car. Regarding your other query, in sentences as simple as this one, it does not make much of a difference whether you use は or が. Since there is only one subject referred to, we can default to は.