Ég tapa leiknum.

Because the verb tapa takes the dative case in Icelandic.

The noun leikur means game. Its forms here are roughly:

• leikur = nominative, a game / game as subject
• leikinn = accusative definite, the game
• leiknum = dative definite, the game after certain verbs/prepositions

Since tapa is one of the verbs that governs the dative, you say:

• Ég tapa leiknum = I lose the game

Even though English just says the game, Icelandic changes the noun form depending on the grammar of the sentence.

Yes, when tapa means to lose, the thing being lost is normally in the dative.

Examples:

• Ég tapa lyklunum. = I lose the keys.
• Hann tapaði peningunum. = He lost the money.
• Við töpum leiknum. = We lose the game.

This is something you mainly have to memorize with the verb:
tapa + dative

Tapa is the 1st person singular present tense form of the verb að tapa.

So:

• ég tapa = I lose
• þú tapar = you lose
• hann/hún/það tapar = he/she/it loses

In the sentence Ég tapa leiknum, the subject is ég (I), so the verb form is tapa.

In Icelandic, the definite article is usually attached to the end of the noun instead of standing as a separate word.

So:

• leikur = a game / game
• leiknum = the game (in the dative singular)

That -num part includes the definite meaning here.

This is very different from English, where the is a separate word.

The dictionary form is leikur.

When you look up nouns in Icelandic dictionaries, you usually look for the nominative singular indefinite form, which is the basic form of the noun.

So:

• sentence form: leiknum
• dictionary form: leikur

It helps to think of leiknum as leikur changed for:

1. dative case
2. singular
3. definite (the)

A simple learner-friendly approximation is:

• Ég ≈ yegg or yeg
• tapa ≈ TAH-pa
• leiknum ≈ LAYK-num

A few pronunciation notes:

• é in Ég sounds like ye.
• a in tapa is broad, like a in father.
• ei in leiknum sounds roughly like ay in day.
• The g in Ég is not always pronounced like a strong English g in all contexts, but for beginners yeg is a helpful approximation.

It can depend on context.

The plain present tense in Icelandic can sometimes correspond to English simple present and sometimes to what English expresses differently. But if you want to make the idea of an ongoing action clearer, Icelandic often uses:

• Ég er að tapa leiknum. = I am losing the game.

So:

• Ég tapa leiknum = I lose the game / sometimes contextually I’m losing the game
• Ég er að tapa leiknum = more explicitly I am losing the game

For a learner, it is safest to understand Ég tapa leiknum first as I lose the game.

The past tense is:

• Ég tapaði leiknum. = I lost the game.

Here, tapaði is the past tense of tapa.

So you can compare:

• Ég tapa leiknum. = I lose the game.
• Ég tapaði leiknum. = I lost the game.

The noun leiknum stays in the dative because tapa still governs the dative in the past tense too.

Usually, no. Icelandic normally keeps the subject pronoun.

So the normal sentence is:

• Ég tapa leiknum.

Unlike some languages, Icelandic is not a language where subject pronouns are regularly dropped. Even though the verb form gives some information, the pronoun is usually still said.

Icelandic follows a verb-second pattern in many main clauses. In a simple sentence like this:

• Ég = first element
• tapa = verb in second position
• leiknum = object

So the order is very natural.

You can also move another element to the front, but the verb still tends to stay second:

• Í dag tapa ég leiknum. = Today I lose the game.

Here Í dag is first, so tapa still comes second, and ég moves after it.

Both can relate to losing, but they are used differently.

• tapa is commonly used for losing a game, battle, competition, money, etc.
• missa often means lose, miss, or fail to catch/keep, depending on context

Examples:

• Ég tapa leiknum. = I lose the game.
• Ég missi af strætónum. = I miss the bus.
• Ég missti lyklana. = I lost the keys.

So for losing a game, tapa is the natural choice.

A yes/no question is usually made by putting the verb first:

• Tapa ég leiknum? = Am I losing the game? / Do I lose the game?

This is a very common Icelandic pattern.

Compare:

• Ég tapa leiknum. = statement
• Tapa ég leiknum? = question

You would remove the definite ending and keep the correct case.

Since tapa takes the dative, a game would be:

• leik? No — not here.
• The correct dative singular indefinite form is leik? Actually for leikur, the dative singular is leik? No, it is leik in some related forms? Wait — the correct form here is leik?
  The correct dative singular indefinite of leikur is leik in modern Icelandic usage? This is a common point of confusion, so the best thing is to learn the noun’s case forms carefully.

For this sentence, the definite form you were given is:

• Ég tapa leiknum. = I lose the game.

If you want to say I lose a game, the correct indefinite dative form is:

• Ég tapa leik.

So the contrast is:

• leik = a game (dative singular indefinite)
• leiknum = the game (dative singular definite)

It is singular.

So:

• leiknum = the game (singular, dative)
• leikjunum = the games (plural, dative)

Example:

• Ég tapa leiknum. = I lose the game.
• Ég tapa leikjunum. = I lose the games.

This is useful because Icelandic endings often show both case and number at the same time.