Planen er vanskelig, dog mulig.

No. In Norwegian, dog in this sentence means however, nevertheless, or though.

So Planen er vanskelig, dog mulig means something like:

• The plan is difficult, however possible
• The plan is difficult, though possible

It is completely unrelated to the English noun dog.

Also, this Norwegian dog is a fairly formal or literary word, especially in modern usage.

Yes, it is grammatical, but it sounds formal and a bit written or literary.

A more everyday version would usually be:

• Planen er vanskelig, men mulig.

You might also hear:

• Planen er vanskelig, men den er mulig.

So the original sentence is correct, but most learners should recognize that dog is not the most common conversational choice.

Because planen is the definite singular form of en plan.

• en plan = a plan
• planen = the plan

So the sentence is talking about the specific plan, not just any plan.

They are predicate adjectives, meaning they come after er and describe planen.

Here, both vanskelig and mulig are in their basic singular form.

A useful detail:

• adjectives ending in -ig often have the same form in common gender singular and neuter singular

So you get:

• en vanskelig plan
• et vanskelig spørsmål
• planen er vanskelig
• spørsmålet er vanskelig

Likewise:

• en mulig løsning
• et mulig svar
• det er mulig

In the plural, these adjectives normally change:

• vanskelige
• mulige

Example:

• Planene er vanskelige, men mulige.

Because Norwegian, like English, often leaves out repeated words when they are easy to understand from context.

The full idea is basically:

• Planen er vanskelig, dog mulig
• understood as: Planen er vanskelig, men den er mulig

So the second half is elliptical: the subject and verb are omitted because they are already understood.

English does this too:

• The plan is difficult, but possible.

You do not need to repeat it is there either.

The comma helps mark the contrast and the pause before the second part.

Here, dog mulig works almost like an inserted contrasting comment: however possible / though possible.

In a sentence like this, the comma gives the sentence a polished written feel:

• Planen er vanskelig, dog mulig.

This punctuation fits the formal style of dog. In more everyday Norwegian, people often avoid this structure entirely and use men instead.

Yes, and that is often the most natural choice for everyday Norwegian.

• Planen er vanskelig, men mulig.

This keeps almost the same meaning, but sounds more modern and ordinary.

There is a slight nuance:

• men = plain but
• dog = more formal however / nevertheless / though

So dog can sound a little more elegant or literary.

It is usually treated more like a sentence adverb or connective adverb than a simple conjunction.

In practical terms, though, the important thing for a learner is this:

• men directly links things as but
• dog adds a contrasting idea more like however / nevertheless

So in meaning, it behaves similarly to a contrast word, but grammatically it is not exactly the same kind of word as men.

Yes, but the sentence becomes even more formal or literary.

For example, in a fuller sentence you might see:

• Planen er dog mulig.

That means something like The plan is, however, possible.

You may also encounter dog later in a clause in older or more formal writing. But for most learners, the main point is:

• recognize dog when reading
• prefer men, likevel, or allikevel when speaking more naturally

It is not pronounced like the English word dog.

A simple learner-friendly approximation is:

• dohg with a long o

The exact pronunciation depends somewhat on dialect, but the key point is that it does not sound like English dog with the short vowel of hot or dog in many English accents.

So when reading it, do not let the English spelling mislead you.