Það er stutt síðan dóttir mín byrjaði að hoppa á trampólíninu.

This is a very common Icelandic time expression. Literally, it is something like It is short since ..., but in natural English it means:

• It has not been long since ...
• It is only a short time since ...

So Það er stutt síðan dóttir mín byrjaði að hoppa á trampólíninu means that only a short amount of time has passed since that event happened.

Here það is not really that. It is a dummy subject, like English it in:

• It is raining
• It is late
• It has not been long since ...

There is no specific thing that það refers to. It is just needed to make the sentence structure work.

Stutt is the neuter singular form of the adjective stuttur meaning short.

In this expression, Icelandic uses the neuter singular form:

• Það er stutt síðan ...

This is standard idiomatic usage. You can think of it as referring to an understood idea like a short time. Because the construction is impersonal and uses það, the adjective appears in the neuter singular form stutt.

Here síðan means since and introduces the clause that tells you the earlier event:

• síðan dóttir mín byrjaði ...
• since my daughter started ...

A useful thing to know is that síðan can mean different things in other contexts, for example later or afterwards, but in this sentence it is a conjunction meaning since.

Because the starting happened at a specific point in the past.

The logic is:

• event: my daughter started jumping
• time passed since that event: not long

So Icelandic naturally says:

• Það er stutt síðan ... byrjaði ...

Even though English may sometimes use a present perfect idea in translation, Icelandic very often uses the simple past in the síðan clause because the event itself is viewed as completed in the past.

Because byrja að + infinitive is the normal pattern for begin/start to do something.

So:

• byrjaði = started
• að hoppa = to jump

Together:

• byrjaði að hoppa = started to jump / started jumping

The word að here is the infinitive marker, like English to.

In Icelandic, possessive pronouns often come after the noun in ordinary, neutral speech:

• dóttir mín = my daughter
• húsið mitt = my house

Putting the possessive first, as in mín dóttir, is possible, but it usually sounds more emphatic, contrastive, or literary.

So in a normal sentence, dóttir mín is the most natural choice.

Because á can take either dative or accusative, and the choice changes the meaning.

Here is the basic idea:

• á + dative = location, position, or activity on something
• á + accusative = motion onto something, toward a destination

So:

• hoppa á trampólíninu = jump on the trampoline as an activity already taking place there
• hoppa á trampólínið = jump onto the trampoline

In your sentence, the meaning is that she started doing the activity of jumping on the trampoline, so dative is used: á trampólíninu.

Trampólíninu is the definite dative singular form of trampólín.

Breakdown:

• trampólín = trampoline
• trampólíni = on a trampoline or to/at a trampoline in dative singular, indefinite
• trampólíninu = the trampoline in dative singular, definite

The ending includes the suffixed definite article, which Icelandic usually attaches to the noun instead of using a separate word like English the.

So:

• á trampólíni = on a trampoline
• á trampólíninu = on the trampoline

Because it follows byrjaði að, the whole phrase is best understood as an activity:

• started jumping on the trampoline

Even though hoppa by itself is an infinitive meaning to jump, after byrja að English often translates it more naturally with -ing:

• byrja að lesa = start reading
• byrja að hlaupa = start running
• byrja að hoppa = start jumping

So byrjaði að hoppa á trampólíninu is most naturally started jumping on the trampoline.

This sentence contains both letters, which are often new for English speakers.

• þ is like the th in thing
• ð is like the th in this

Examples from the sentence:

• Það begins with þ, so the first sound is like th in thing
• dóttir contains tt, not ð
• byrjaði contains ð, pronounced like the th in this

A rough guide:

• Það ≈ thath with the thing sound at the start
• byrjaði ≈ bir-ya-thi with a voiced th

The exact pronunciation is more subtle than English spelling suggests, but that is a good starting point.

Yes, that is a very helpful way to understand it.

The sentence has:

1. a time-passed expression

   • Það er stutt síðan ...
   • It has not been long since ...

2. the event that happened earlier

   • dóttir mín byrjaði að hoppa á trampólíninu
   • my daughter started jumping on the trampoline

So the full structure is:

• [It has not been long since] + [earlier event]

That pattern is extremely useful in Icelandic, and once you know it, you can make many similar sentences.