Feluleikurinn er skemmtilegri í garðinum en inni.

It is a compound noun plus the definite article:

• feluleikur = hide-and-seek / hiding game
• -inn = the definite article the

So feluleikurinn means the hide-and-seek game or hide-and-seek as a specific game being talked about.

The ending looks like -urinn because the base noun is feluleikur, a masculine noun ending in -ur. In the nominative singular definite form, that becomes -urinn.

Icelandic and English do not use definiteness in exactly the same way.

Here, feluleikurinn treats the game as a known or identifiable activity: the game of hide-and-seek. Icelandic can sound natural with the definite form in places where English would simply use the name of an activity with no article.

So even if English says just hide-and-seek, Icelandic may still use feluleikurinn.

Because the sentence is making a comparison: the game is more fun in one place than in another.

• skemmtilegur = fun, entertaining
• skemmtilegri = more fun

This is the comparative form of the adjective. It also matches the subject feluleikurinn, which is masculine singular here.

Because í can take different cases depending on meaning:

• dative for location: in
• accusative for movement into something: into

Here the meaning is location, so Icelandic uses the dative:

• í garðinum = in the garden

If you said í garðinn, that would mean movement into the garden.

Yes, it could, but the meaning would change slightly.

• í garðinum = in the garden (a specific garden)
• í garði = in a garden (not a specific one)

Icelandic shows definiteness by attaching the article to the noun, instead of using a separate word like English the.

Because inni is an adverb, not a preposition plus noun.

It means inside / indoors, and it can stand alone. English does the same with indoors.

So:

• í garðinum = preposition + noun phrase = in the garden
• inni = adverb = indoors / inside

If you wanted to name the exact place, you could use a fuller phrase such as inni í húsinu = inside the house.

Here en means than.

It is used after a comparative adjective:

• skemmtilegri ... en ... = more fun ... than ...

Be aware that en can also mean but in other sentences, but after a comparative like skemmtilegri, it is understood as than.

Yes. Icelandic word order is fairly flexible, but it follows the verb-second rule in main clauses.

The original sentence is a neutral way to say it:

• Feluleikurinn er skemmtilegri í garðinum en inni.

You could also front the location for emphasis:

• Í garðinum er feluleikurinn skemmtilegri en inni.

That puts more focus on in the garden, but the verb er still stays in second position.