No estoy de acuerdo con esa idea.

Because Spanish uses the fixed expression estar de acuerdo for to agree / to be in agreement.

So you say:

• Estoy de acuerdo = I agree
• No estoy de acuerdo = I don’t agree

Even though agreement can feel like a more permanent opinion in English, Spanish still uses estar, not ser, in this expression. Saying soy de acuerdo is not correct.

De acuerdo is a set phrase meaning in agreement.

So:

• estar de acuerdo = to agree
• no estar de acuerdo = to disagree

It’s best to learn it as a chunk, not word by word.

Also, de acuerdo can mean okay / all right in other contexts:

• —Vamos a las seis. —De acuerdo.
  —Let’s go at six. —Okay.

But in your sentence, it clearly means in agreement.

Because in Spanish you are de acuerdo con someone or something.

The pattern is:

estar de acuerdo con + person / thing / idea

Examples:

• Estoy de acuerdo con Ana. = I agree with Ana.
• Estoy de acuerdo con eso. = I agree with that.
• No estoy de acuerdo con esa idea. = I don’t agree with that idea.

English also uses with after agree, so this part lines up quite nicely.

In standard Spanish, negation usually goes directly before the conjugated verb:

• No estoy... = I am not...
• No tengo... = I do not have...
• No quiero... = I do not want...

So No estoy de acuerdo is the normal way to say I do not agree.

You would not say Estoy no de acuerdo.

Because Spanish often drops subject pronouns when the verb already shows who is doing the action.

Here, estoy already tells you it means I am, so yo is unnecessary.

• No estoy de acuerdo... = normal, natural
• Yo no estoy de acuerdo... = also correct, but more emphatic

You might add yo if you want contrast or emphasis:

• Ellos sí, pero yo no estoy de acuerdo.
  They do, but I don’t agree.

Esa means that, while esta means this.

Very roughly:

• esta idea = this idea
• esa idea = that idea
• aquella idea = that idea over there / that more distant idea

In conversation, esa idea often refers to an idea already mentioned by someone else, or something a little more removed from the speaker.

So:

• No estoy de acuerdo con esta idea. = I don’t agree with this idea.
• No estoy de acuerdo con esa idea. = I don’t agree with that idea.

Because idea is a feminine noun:

• la idea
• esa idea

Adjectives and demonstratives have to match the noun’s gender and number:

• ese libro = that book
• esa idea = that idea

So ese idea would be incorrect.

Yes. If the context is already clear, No estoy de acuerdo sounds completely natural.

For example:

• —¿Te parece buena propuesta? —No estoy de acuerdo.
  —Do you think it’s a good proposal? —I don’t agree.

You only add con esa idea when you want to specify what you disagree with.

Yes. The same structure works with people, opinions, plans, statements, and more.

Examples:

• No estoy de acuerdo con María. = I don’t agree with María.
• No estoy de acuerdo contigo. = I don’t agree with you.
• No estoy de acuerdo con lo que dices. = I don’t agree with what you’re saying.

So estar de acuerdo con is very flexible.

No. In this sentence, acuerdo is part of the fixed expression de acuerdo, where it functions like a noun phrase meaning agreement.

That’s why the whole thing works as:

• estar de acuerdo = to be in agreement

This can confuse learners because acuerdo also exists as a verb form in other contexts:

• me acuerdo = I remember

But in No estoy de acuerdo, it does not mean I remember. It means I am not in agreement.

Yes, that is also correct.

• No estoy de acuerdo con esa idea.
• Estoy en desacuerdo con esa idea.

Both mean I don’t agree with that idea.

The difference is mainly style:

• No estoy de acuerdo is very common and neutral.
• Estoy en desacuerdo sounds a bit more formal or explicit.

For everyday conversation, No estoy de acuerdo is usually the more natural choice.

A simple approximation is:

no es-TOY de a-KWER-do con E-sa i-DE-a

A few helpful notes:

• estoy has the stress on -toy
• acuerdo has the stress on -cuer-
• idea has the stress on de

In Spain, a careful pronunciation would be roughly:

no es-TOY de a-KWER-do kon E-sa i-DE-a

If you want to sound natural, say it as a smooth phrase rather than word by word:

Noestoydeacuerdoconesaidea.

No, it’s generally neutral and polite. It simply states disagreement.

• No estoy de acuerdo con esa idea. = clear, direct, normal

If you want to sound softer, you could say:

• No estoy muy de acuerdo con esa idea. = I’m not really in agreement with that idea.
• No sé si estoy de acuerdo con esa idea. = I’m not sure I agree with that idea.
• No creo que esa idea sea buena. = I don’t think that idea is good.

So the original sentence is direct, but not rude by itself.