Ég vil ekki missa af leiknum í kvöld.

Because vil is the 1st person singular present form of the verb vilja (to want).

So the verb changes like this:

• ég vil = I want
• þú vilt = you want
• hann/hún/það vill = he/she/it wants

The dictionary form is vilja, but after ég you need the conjugated form vil.

Because vilja is followed by a bare infinitive in Icelandic.

So:

• Ég vil missa af leiknum. = I want to miss the game.
• Ég vil ekki missa af leiknum. = I do not want to miss the game.

You do not say ég vil að missa here.

Many Icelandic verbs behave this way, especially modal-like verbs such as vilja.

In a normal main clause, the finite verb usually comes early in the sentence, and ekki typically comes after that finite verb.

So:

• Ég vil ekki... = I do not want...
• not Ég ekki vil...

This is very normal Icelandic word order.

A helpful way to see the structure is:

• Ég = subject
• vil = finite verb
• ekki = negation
• missa af leiknum í kvöld = rest of the verb phrase

Missa af is the normal expression for to miss in the sense of fail to catch / fail to attend / not experience something.

So:

• missa af strætónum = miss the bus
• missa af leiknum = miss the game
• missa af tækifærinu = miss the opportunity

The preposition af is part of the expression here. You should learn missa af as a unit.

This is different from plain missa, which often means lose something:

• missa lyklana = lose the keys

So the af is important.

Because af takes the dative case, and leiknum is the dative singular definite form of leikur (game).

Forms of leikur:

• leikur = a game (nominative singular)
• leikinn = the game (accusative singular, often depending on role)
• leiknum = the game (dative singular)

Since the phrase is missa af + dative, you get:

• af leiknum = the game

So in this sentence, leiknum is exactly the form you need.

Here, -num shows two things at once:

• dative case
• definite article (the)

So leiknum means the game in the dative singular.

This is very common in Icelandic: the definite article is usually attached to the noun rather than written as a separate word.

Compare:

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

Because í kvöld is a fixed idiomatic time expression meaning tonight / this evening.

Icelandic often uses simple prepositional time phrases like this without the definite article:

• í dag = today
• í morgun = this morning
• í kvöld = tonight

So í kvöld is just the normal way to say it.

It is a time expression telling you when the action is relevant.

So the sentence breaks down roughly like this:

• Ég = I
• vil ekki = do not want
• missa af leiknum = miss the game
• í kvöld = tonight

It modifies the whole situation: the speaker does not want to miss the game tonight.

Yes, Icelandic allows some flexibility, especially if you want to emphasize time or another element.

For example:

• Ég vil ekki missa af leiknum í kvöld.
• Í kvöld vil ég ekki missa af leiknum.

Both are grammatical. The second one emphasizes tonight more.

Notice what happens in the second sentence: when Í kvöld comes first, the finite verb vil still stays in the second position, and ég moves after it. This is a very important Icelandic pattern.

A rough learner-friendly guide would be:

yeg vil ehk-ki missa av layk-num ee kvvöld

A few useful notes:

• Ég begins with a sound somewhat like yehg for English speakers.
• ll in kvöld does not sound like a normal English l.
• af is often pronounced more like av in connected speech.
• leiknum has the diphthong ei, roughly like layk.

If you want to sound more natural, it is best to listen to native audio, because Icelandic spelling does not always map neatly onto English-style pronunciation.

No, not naturally.

Because of missa af, the meaning is to miss out on / fail to attend or catch the game, not to lose the game as a participant.

So:

• Ég vil ekki missa af leiknum í kvöld. = I don’t want to miss the game tonight.

If you wanted to say lose the game, you would use a different verb, such as tapa:

• Ég vil ekki tapa leiknum í kvöld. = I don’t want to lose the game tonight.

So missa af and tapa are very different.