Im Sommer esse ich Mango lieber kalt.

Im is a contraction of in dem.

With seasons used as time expressions, German normally says:

• im Sommer
• im Winter
• im Herbst
• im Frühling

So im Sommer means in the summer / during summer.

This is a very common fixed pattern. Saying in Sommer would be ungrammatical here.

Because German main clauses follow the verb-second rule.

The first position in the sentence is taken by Im Sommer. Once that happens, the conjugated verb must come next:

• Im Sommer | esse | ich | Mango lieber kalt.

So the subject ich moves after the verb.

Compare:

• Ich esse im Sommer Mango lieber kalt.
• Im Sommer esse ich Mango lieber kalt.

Both are grammatical, but when Im Sommer comes first, esse must stay in second position.

In German, foods are often used without an article when you mean them in general as a kind of food:

• Ich esse Mango.
• Ich trinke Kaffee.
• Wir essen Pizza.

So Mango here means something like mango as food, not necessarily one whole mango.

If you wanted to emphasize one fruit, you would usually say:

• Ich esse eine Mango.

That sounds more like I am eating a mango or one mango.

So the version without an article is natural when talking about general preference.

Lieber means preferably / rather / more willingly. It expresses a preference.

In this sentence, it means the speaker prefers mango cold.

A useful comparison:

• gern = gladly / likes doing
• lieber = rather / prefers
• am liebsten = most of all

Examples:

• Ich esse Mango gern. = I like eating mango.
• Ich esse Mango lieber kalt. = I prefer eating mango cold.
• Ich esse Mango am liebsten kalt. = I like eating mango cold best / most of all.

So lieber is the comparative form used for preference.

Here kalt is not directly in front of the noun Mango, so it is not an attributive adjective.

It is describing how the mango is eaten: cold.

German often uses the plain adjective form in this kind of adverb-like function:

• Ich trinke Kaffee schwarz.
• Sie isst Fisch roh.
• Wir essen Suppe heiß.

So kalt stays just kalt.

It comes at the end because that is a very natural place in German for this kind of descriptive element, especially after lieber.

Yes, but the structure changes slightly.

• Mango kalt essen = to eat mango cold
• kalte Mango = cold mango

So:

• Im Sommer esse ich Mango lieber kalt.
  This focuses on the preference for eating it cold.

• Im Sommer esse ich lieber kalte Mango.
  This treats cold mango as a noun phrase.

Both can work, but they are not exactly the same in structure.

Also notice that when the adjective comes before the noun, it needs an ending:

• kalte Mango

That is because it is now directly modifying the noun.

Yes. German word order is flexible, but the emphasis changes.

For example:

• Im Sommer esse ich Mango lieber kalt.
• Ich esse im Sommer Mango lieber kalt.
• Mango esse ich im Sommer lieber kalt.

All of these can be grammatical in the right context.

What stays important is the verb-second rule in a main clause: whatever comes first, the conjugated verb still has to be second.

So if Mango is moved to the front for emphasis, you get:

• Mango esse ich im Sommer lieber kalt.

That sounds more marked, as if you are contrasting mango with something else.

Yes, it is natural and understandable.

It sounds like a general statement about preference: in summer, the speaker prefers mango cold.

Depending on context, a native speaker might also say things like:

• Im Sommer esse ich Mango am liebsten kalt.
• Im Sommer esse ich lieber kalte Mango.
• Ich esse Mango im Sommer lieber kalt.

The original sentence is perfectly fine; these are just stylistic alternatives.