Translation:I do not want a pink shirt.
The negative modifier "ない" itself is an adjective. When you precede another i-adjective before it, you need to convert it into an adverb first, so "欲しい" being an i-adjective becomes "ほしく" an adverb before "ない". This is a different story for na-adjective, e.g. not好きな -> 好きではない (wrong: 好きにない). And no, a ku-adverb should not be treated as a verb whatsoever, it just happens to look like a ku-verb, but the inflections are completely different.
Not really because PINKU could be a person, then if there's no context you won't be able to tell whether it's "Pink's shirt" or "a pink shirt". Later or through different means you'll learn about a particle called "NA" な, which is used to tie adjectives to nouns for descriptions. "Pink na shatsu".
No, cherry blossoms are another color 桜色 さくらいろ which is closer to white. 桃色 ももいろ describes peach blossom color. Most cherry blossoms are almost white with a nuance of red, while most peach blossoms are a more intense color. ピンク can be said to both. But 桃色 can only used for rare, color intense cherry blossoms while most are 桜色 and this is where the word 桜色 even comes from, since 桜 is the cherry blossom. Also, many pictures of Japanese cherry blossoms are photoshopped to make the color more intense. Go see it in real, and you know how it is almost white.
It just makes the expression a noun i.e. "pinku iro no shatsu" is/can be considered a noun. It's the difference between saying "pink shirts" in English,which is a noun, and saying "the shirt is pink." The kanji you used is "ao" which means blue. Red is 赤い. I think the "no" is optional depending on the actual color. I remeber somebody posting about this on the reverse English for Japanese course.
In japanese adjetives are split into 2 categories. And have different rules on how you use them. For example when attaching them to a noun it either goes いadj + noun or なadj + な + noun. Now the reason there is a の is because colors like orange, pink, and green are treated as nouns (dont ask me why it doesnt make sense to me either)
ほしい (hoshii) is an adjective in Japanese, and adjectives are followed by です.
シャツは青いです。 (shatsu wa aoi desu)
The shirt is blue.
シャツが欲しいです。 (shatsu ga hoshii desu)
I want a shirt (literally: a shirt is wanted).
シャツは青くないです。(shatsu wa aokunai desu)
The shirt is not blue.
シャツは欲しくないです。 (shatsu ha hoshikunai desu)
I don't want a shirt (literally: the shirt is not wanted).
The structure is [adjective]です.