Learn Hardcore Japanese: tyuui ga hituyou desu.

注意 (chūi): a noun meaning “attention” or “caution.”
が (ga): the subject-marking particle, indicating that “caution” is what’s necessary.
必要 (hitsuyō): a na-adjective meaning “necessary.”
です (desu): the polite copula, turning “必要” into a polite predicate.

So literally it’s “Caution (が) necessary (です).”

Why is the particle が used after 注意 instead of は?

The pattern Xが必要だ/です uses が to present X as the thing whose existence is required.

が simply marks “caution” as the subject: “It is caution that is necessary.”
If you used は, you’d be making “caution” the topic and might imply contrast (e.g. “As for caution, it’s necessary—but something else may not be”).

What part of speech is 必要, and why does です follow it without another particle?

必要 is a na-adjective (形容動詞). When used predicatively:

In plain speech: 必要だ
In polite speech: 必要です
No extra particle is needed between a na-adjective and だ/です, just like 静かだ or 元気です.

How do I turn 注意が必要です into plain form, negative, or past tense?

Here are some common variations:

Plain affirmative:

注意が必要だ

Polite negative:

注意が必要ではありません
注意が必要じゃありません

Plain negative:

注意が必要ではない
注意が必要じゃない

Polite past affirmative:

注意が必要でした

Polite past negative:

注意が必要ではありませんでした

If I want to say “you need to pay attention” using a verb, is 注意する必要があります correct?

Yes. 注意する is the verb “to pay attention” or “to be careful.” The construction ～する必要がある means “there is a need to do ～.”
So 注意する必要がある literally means “there is a need to pay attention,” which is very natural.

What’s the nuance difference between 注意が必要だ and 注意する必要がある?

注意が必要だ/です treats 注意 as a noun: “Caution is necessary.” It’s slightly more abstract.
注意する必要がある treats 注意する as a verb: “There is a need to take care/pay attention.” It focuses on the action you must perform.

Can I say 注意が要る (注意がいる) instead of 注意が必要だ?

No, 要る (“to need,” usually いる in kana) is used for concrete items (e.g. お金が要る “I need money”). It’s unnatural to say 注意が要る. Use 必要だ/です for more abstract needs like “caution,” “time,” “permission,” etc.