Min nabo siger, at hun ville ønske, at hun havde bedre kondition, men hun træner stadig hver uge.

Because Danish normally uses the basic noun form after a possessive like min, din, hendes, etc.

• min nabo = my neighbour
• naboen = the neighbour

So you do not combine min with the definite ending -en here. That is different from English, where the noun form does not change.

By itself, nabo just means neighbour. It is not specifically male.

In this sentence, we know the neighbour is female because of hun:

• hun = she

So:

• Min nabo = My neighbour
• hun tells you that this neighbour is a woman

Each at introduces a new clause.

1. Min nabo siger, at ...
   Here at means that after siger (says).

2. hun ville ønske, at ...
   Here at introduces the content of the wish.

So the structure is:

• My neighbour says [that ...]
• inside that clause: she wishes [that ...]

In natural Danish, the second at is often omitted in speech:

• Min nabo siger, at hun ville ønske, hun havde bedre kondition ...

That version is also very common.

Ville ønske is a very common way to express an unreal or contrary-to-fact wish about the present.

• hun ønsker bedre kondition = she wants better fitness
• hun ville ønske, at hun havde bedre kondition = she wishes she had better fitness

The second version sounds more like:

• this is not true now
• she feels a lack
• she wishes reality were different

So ville ønske works a lot like English would wish / wishes she had.

This is one of the most important patterns in the sentence.

After ville ønske, Danish often uses a past tense form to express an unreal present situation.

So:

• hun havde bedre kondition does not mean the wish is about the past
• it means she does not have better fitness now, but wishes she did

This is very similar to English:

• I wish I had more time
• She wishes she were taller
• She wishes she had better fitness

So the past form here marks distance from reality, not past time.

Not really. Kondition in Danish usually means physical fitness, especially stamina or cardio fitness.

So in this sentence:

• bedre kondition = better fitness / better stamina

It does not usually mean general condition in the broad English sense.

A useful warning:

• English condition can mean health state, situation, requirement, etc.
• Danish kondition is much narrower and is strongly associated with exercise and fitness

Because Danish often leaves out the article with abstract nouns, especially in phrases like this.

• have bedre kondition = have better fitness
• få bedre kondition = get better fitness

That is the most natural, idiomatic form here.

You may sometimes hear en bedre kondition, but bedre kondition is the more neutral and common phrasing in this kind of sentence.

Stadig means still.

So:

• men hun træner stadig hver uge = but she still works out every week

It shows contrast:

• she is not satisfied with her fitness
• but she continues to train regularly

Because this is a main clause, and Danish main clauses normally have the finite verb in second position.

So:

• hun træner stadig hver uge

Here:

• hun = subject
• træner = finite verb
• stadig = adverb

This order is very natural in a main clause.

A useful comparison:

• Hun træner stadig hver uge.
  main clause

• ... at hun stadig træner hver uge.
  subordinate clause

So adverb placement often changes depending on whether the clause is a main clause or an at-clause.

Because hver uge simply means every week, and Danish does not need a preposition here.

Examples:

• hver dag = every day
• hver uge = every week
• hver måned = every month

This works very much like English.

They are a mix of forms, and each has a different job:

• siger = present tense, says
• ville = formally a past form of vil, but here it helps express an unreal wish
• havde = past form, used here for an unreal present situation
• træner = present tense, works out / trains

So even though ville and havde look past, the overall meaning is still about the present:

• she says this now
• she wishes now
• she trains now

Not always. This depends on which Danish comma system is being used.

In the sentence you were given, the writer uses commas before the subordinate clauses:

• Min nabo siger, at hun ville ønske, at hun havde bedre kondition, men hun træner stadig hver uge.

In modern Danish, many writers omit the comma before at:

• Min nabo siger at hun ville ønske at hun havde bedre kondition, men hun træner stadig hver uge.

But the comma before men is standard and should stay.

So the short answer is:

• before men: yes
• before at: depends on comma style

Yes, absolutely.

• bedre kondition focuses more on stamina / cardio fitness
• i bedre form is broader and can mean in better shape overall

So these are both natural, but they are not exactly identical.

Compare:

• hun havde bedre kondition = she had better endurance/fitness
• hun var i bedre form = she was in better shape overall

In everyday speech, i bedre form is very common.