I'm a Japanese student studying English at university. Let me try and answer your question.
We use 特殊 tokushu in academic situations, describing something strange or crazy.
In contrast, we say 特別 tokubetsu in casual situation, describing something precious or valuable.
For example, we say 「君は特別な人だ」(kimi wa tokubetsu na hito da - "You are precious to me"), but we don't say 「君は特殊な人だ」(kimi wa tokushu na hito da) in that sense.
When we say 「君は特殊な人だ」, this means "You are strange" or "You are not an ordinary person."
The answer for your next question is clear:
「特別な」and「特殊な」are correct, and they are not noun but adjective that can be translated as "special" or "peculiar."
We don't say 「特別の」or 「特殊の」in Japanese.
What's more, to be precise, "special" is an adjective (形容詞 keiyōshi) in English, but 「特別な」「特殊な」is 「形容動詞」(keiyōdōshi) in Japanese.
「「形容動詞」can be classified as adjective, but we distinguish 「形容詞」