Jeg er uenig med min ven om planen.

Because Danish uses a very common pattern:

uenig med + person + om + topic

So here:

• med min ven = the person I disagree with
• om planen = the thing we disagree about

A useful model is:

• Jeg er uenig med ham om det. = I disagree with him about it.
• Hun er uenig med sin lærer om svaret. = She disagrees with her teacher about the answer.

So med marks the other person, and om marks the subject of the disagreement.

Yes. Both are natural, and both can mean I disagree.

There is often little or no difference in everyday use:

• Jeg er uenig.
• Jeg er ikke enig.

Both are common.

A small nuance:

• uenig is a single adjective meaning in disagreement / not in agreement
• ikke enig is the negated form of enig

In many situations, they are interchangeable. In this sentence, Jeg er uenig med min ven om planen is perfectly natural.

In Danish, possessives like min, din, hans, hendes, vores, and so on normally come before the noun, and the noun does not take the definite ending.

So:

• min ven = my friend
• din bil = your car
• hans hus = his house

You do not say:

• min vennen
• min venen
• minen ven

The possessive already makes the noun specific, so no extra definite marking is needed.

Because Danish usually puts definiteness at the end of the noun.

• en plan = a plan
• planen = the plan

This -en ending is the common definite ending for many common-gender nouns.

So in the sentence, planen means the plan, not just a plan.

Compare:

• Vi har en plan. = We have a plan.
• Planen er god. = The plan is good.

Literally, yes, med often corresponds to with, but in this sentence it does not mean being together with your friend in a physical sense.

Here, med is part of the pattern used with uenig:

• uenig med nogen = disagree with someone

So med min ven means with my friend in the sense of in relation to my friend / against my friend's view, not alongside my friend.

That would usually sound wrong or at least awkward if you mean I disagree with my friend about the plan.

The normal order is:

uenig med nogen om noget

So the best version is:

Jeg er uenig med min ven om planen.

If you move the phrases around, it may become unclear whether med min ven belongs to uenig or to planen.

Uenig is an adjective.

It is built from:

• enig = agreed / in agreement
• u- = a negative prefix, like un- in English

So:

• enig = in agreement
• uenig = not in agreement / in disagreement

In this sentence, it comes after er and describes the subject jeg:

• Jeg er uenig = I am in disagreement / I disagree

Because Danish does not need a helper verb like do in this kind of sentence.

English often says:

• I disagree
• I do not agree

Danish commonly uses:

• Jeg er uenig.
• Jeg er ikke enig.

So instead of using do, Danish often uses to be + adjective:

• er uenig
• er enig

This is very normal Danish.

Yes. In this sentence, om is very close to about.

So:

• om planen = about the plan

But it is best to learn it as part of the whole pattern:

uenig med nogen om noget

That will help you sound natural, instead of translating each word one by one.

Yes, if the context is already clear.

For example:

• Jeg er uenig med min ven.
  = I disagree with my friend.
  The topic is not stated.

• Jeg er uenig om planen.
  This is less complete by itself and can sound a bit odd unless the people involved are understood from context.

Most often, if you want the full idea, Danish uses both:

Jeg er uenig med min ven om planen.

That is the clearest full version.

In careful writing it is jeg, but in everyday spoken Danish the pronunciation is often reduced.

Very broadly:

• jeg is often pronounced something like yai or ya
• er is often weak in fast speech
• uenig has the stress on the last part: u-e-NI

So in natural speech, the whole sentence may sound much smoother and less clearly pronounced than the spelling suggests.

This is normal in Danish: spoken forms are often more reduced than written forms.

It is neutral and completely natural in both speech and writing.

It is not especially formal, and it is not slangy either. You could use it:

• in conversation
• in writing
• in class
• at work

If you wanted to sound a little softer or more diplomatic, you might say something like:

• Jeg er ikke helt enig med min ven om planen. = I do not completely agree with my friend about the plan.
• Jeg ser lidt anderledes på planen end min ven. = I see the plan a bit differently from my friend.

But the original sentence is perfectly normal.