Það eru tvær vikur síðan hún byrjaði að æfa fótbolta í nýja íþróttahúsinu.

This is a very common Icelandic pattern for talking about how much time has passed since something happened:

Það er/eru + time period + síðan + clause

So:

Það eru tvær vikur síðan...
means
It has been two weeks since...

Here það works like a dummy subject, similar to English it in It has been...

Because tvær vikur is plural.

Icelandic verbs agree with the noun phrase here:

• Það er ein vika síðan... = It has been one week since...
• Það eru tvær vikur síðan... = It has been two weeks since...

So the plural noun phrase tvær vikur triggers plural eru.

Because vika is a feminine noun, and the word for two changes according to gender:

• tveir for masculine nouns
• tvær for feminine nouns
• tvö for neuter nouns

Examples:

• tveir dagar = two days
• tvær vikur = two weeks
• tvö ár = two years

Since vika is feminine, the correct form is tvær.

In this construction, the time expression appears as nominative plural, so you get tvær vikur.

You would see vikum in a different structure, for example after a preposition:

• fyrir tveimur vikum = two weeks ago

So compare:

• Það eru tvær vikur síðan... = It has been two weeks since...
• fyrir tveimur vikum = two weeks ago

Both are correct, but they belong to different grammatical patterns.

Here síðan means since and introduces the event that happened in the past:

Það eru tvær vikur síðan hún byrjaði...
= It has been two weeks since she started...

It marks the starting point from which the two weeks are being counted.

Be aware that síðan can also mean then or after that in other contexts, so its exact meaning depends on the sentence.

Because síðan introduces a subordinate clause, and subordinate clauses usually have more straightforward word order in Icelandic:

subject + verb

So:

• síðan hún byrjaði = since she started

In main clauses, Icelandic often follows the V2 rule, where the finite verb comes early. But after a conjunction or subordinator like síðan, the normal order is usually used.

Because the act of starting happened at one point in the past and is complete.

The sentence is measuring time from that past event up to the present:

• she started
• two weeks have passed since then

So byrjaði is exactly what you would expect.

If you wanted to focus on the ongoing activity instead, Icelandic would often use a different structure, for example:

Hún hefur æft fótbolta í tvær vikur.
= She has been practicing football for two weeks.

It is the infinitive marker, equivalent to English to.

After byrja when another verb follows, Icelandic uses:

byrja að + infinitive

So:

• byrja að æfa = start to practice
• byrja að læra = start to study
• byrja að vinna = start to work

Æfa literally means practice, train, or rehearse.

With a sport, æfa fótbolta usually means to practice or train in football, often as a regular activity. In natural English, that may simply be translated as play football, depending on context.

You could also hear:

spila fótbolta = play football

The difference is roughly:

• æfa fótbolta = practice/train football
• spila fótbolta = play football

In this sentence, æfa fits well because it suggests organized practice.

Because æfa takes a direct object, and that object is in the accusative case.

The dictionary form is:

fótbolti = football

But after æfa, it becomes accusative singular:

æfa fótbolta

So this is a normal case change caused by the verb.

There are three things happening here:

1. í means in
2. With location, í takes the dative
3. íþróttahúsinu is definite, meaning the sports hall

So:

• íþróttahúsinu = in the sports hall
• the ending -inu is the suffixed definite article

The adjective nýja is also affected by this. Because the noun is definite, the adjective takes the weak form:

• nýja íþróttahúsinu = the new sports hall, in the dative

So the whole phrase means:

í nýja íþróttahúsinu = in the new sports hall

Because adjectives in Icelandic change according to:

• gender
• number
• case
• definiteness

Here the noun is:

• neuter singular: íþróttahús
• dative because of í
• definite because of -inu

That combination requires the weak adjective form nýja.

So:

• nýtt íþróttahús = a new sports hall
• í nýja íþróttahúsinu = in the new sports hall

Yes. A very common alternative is:

Hún byrjaði að æfa fótbolta í nýja íþróttahúsinu fyrir tveimur vikum.

That means:

She started practicing football in the new sports hall two weeks ago.

The difference is mainly one of perspective:

• Það eru tvær vikur síðan... focuses on the time that has passed up to now
• fyrir tveimur vikum simply places the event two weeks in the past

Both are natural.