Das Problem ist mit Ruhe und guter Planung lösbar.

The sentence breaks down like this:

• Das Problem = the subject
• ist = the finite verb
• mit Ruhe und guter Planung = a prepositional phrase
• lösbar = a predicate adjective

So the core is:

• Das Problem ist lösbar. = The problem is solvable.

Then German adds mit Ruhe und guter Planung to show the means or circumstances.

lösbar is an adjective meaning solvable.

It comes from:

• lösen = to solve
• -bar = a suffix meaning something like -able or can be ...

So:

• lösen → lösbar
• compare:
  • machen → machbar = doable
  • essen → essbar = edible

In this sentence, lösbar is used after ist, so it works like an adjective in The problem is solvable.

Because German often places descriptive information like this later in the sentence, especially when there is a prepositional phrase in between.

The core idea is still:

• Das Problem ist lösbar.

German then inserts:

• mit Ruhe und guter Planung

So you get:

• Das Problem ist mit Ruhe und guter Planung lösbar.

This sounds natural in German. The important thing is that ist is still in the second position, which is the normal rule for a main clause.

Because mit always takes the dative case.

So after mit, both nouns are in the dative:

• mit Ruhe
• mit guter Planung

Now look at Planung:

• Planung is feminine
• it appears without an article
• feminine singular dative with no article takes the strong adjective ending -er

So:

• gute Planung = nominative/accusative style form
• guter Planung = dative form after mit

That is why guter is correct here.

Yes. Mit is always followed by the dative.

That is one of the standard German prepositions that always require a fixed case.

So in this sentence:

• mit Ruhe
• mit guter Planung

both are dative.

A useful detail: Ruhe does not visibly change here, because many feminine nouns look the same in nominative, accusative, and dative singular. But the adjective in guter Planung clearly shows the dative.

Because German often leaves out the article with abstract nouns when speaking in a general way.

Here:

• Ruhe means calmness/composure
• Planung means planning

The sentence is talking about these ideas generally, not about one specific calmness or one specific plan.

So:

• mit Ruhe und guter Planung = with calm and good planning

If you added articles, the meaning would become more specific or stylistically different, for example:

• mit der nötigen Ruhe und einer guten Planung

That sounds more like with the necessary calm and a good plan/planning approach.

Here mit Ruhe means something like:

• with calm
• with composure
• by staying calm

It does not mainly mean literal physical quietness here.

Ruhe can mean different things depending on context:

• quiet/silence
• rest
• calmness

In this sentence, it clearly means calmness/composure.

Not in the same way.

mit Ruhe and in Ruhe are related, but they are not the same expression.

• mit Ruhe = with calmness / by remaining calm
• in Ruhe = in peace, undisturbed, calmly, without interruption

For example:

• Lass mich in Ruhe. = Leave me alone.
• Wir sollten das in Ruhe besprechen. = We should discuss that calmly / without rushing.

In your sentence, mit Ruhe is the better fit because it expresses a quality or approach used to solve the problem.

Yes. That is a very natural alternative.

Compare:

• Das Problem ist lösbar.
• Das Problem kann gelöst werden.

They are very similar, but the style is a little different:

• ist lösbar describes the problem as having the property solvable
• kann gelöst werden uses a passive construction and focuses more on the action can be solved

So:

• Das Problem ist mit Ruhe und guter Planung lösbar.
• Das Problem kann mit Ruhe und guter Planung gelöst werden.

Both work well.

Because it is being used as a predicate adjective, not directly before a noun.

Compare:

• ein lösbares Problem = a solvable problem
  
  • here the adjective is before the noun, so it takes an ending
• Das Problem ist lösbar. = the problem is solvable
  
  • here it comes after ist, so it does not take an ending

That is the normal pattern in German:

• before a noun → adjective ending needed
• after sein, werden, bleiben, etc. → usually no ending

Yes, German allows some flexibility.

For example:

• Das Problem ist mit Ruhe und guter Planung lösbar.
• Mit Ruhe und guter Planung ist das Problem lösbar.

Both are correct.

The second version puts more emphasis on mit Ruhe und guter Planung. The original version sounds very neutral and natural.

What usually stays the same in a normal main clause is that the finite verb, here ist, remains in second position.