About Japanese adjectives?

Hello everyone.

I was wondering if anyone could help and answer the few questions I have about Japanese adjectives.

1. Is it true you can drop the "desu" with adjectives that end in "-i", e.g. Ureshii desu (I'm happy), becomes just Ureshii?

Is desu just there to show politeness when talking?

2. Do you always have to use "na" to connect a "-na" adjective to a noun, e.g.: shizuka na onnanoko (the quiet girl)

Even though "shizuka" and not "shizuke na" is the adjective.

3. How do you use more than two adjectives to describe a noun, e.g. a big, red car.

Would it be okii akai kuruma? If not how would you say this in Japanese?

4. Finally could you check these sentences to see whether they're correct or not?

1. Adjectives ending in "-i" (samui, atsui, utsukushii, akai, etc.)

a. - the red car - akai kuruma

b- the car is red - kuruma wa akai desu

2. Adjectives ending in "-na" (kirei na, teinei na, benri na)

a. the rude boy - shitsurei na otokonoko

b. the boy is rude - otokonoko wa shitsurei desu

I would be very grateful if you could help me out so I can continue my Japanese studies without any doubts, thank you for your time =)!