Ich winke ihr zurück und umarme sie später vor dem Haus.

Because the two verbs take different kinds of objects.

• jemandem winken uses the dative
• jemanden umarmen uses the accusative

So:

• ihr = dative her / to her
• sie = accusative her

English uses her for both, but German changes the pronoun depending on its grammatical role.

Because zurückwinken is a separable verb.

In the dictionary form, it is one word:

• zurückwinken

But in a normal main clause, the conjugated verb goes to position 2, and the separable prefix moves to the end:

• Ich winke ihr zurück.

That is completely normal for separable verbs in German.

Other examples:

• Ich stehe auf.
• Er kommt an.
• Wir rufen dich zurück.

Because German winken normally takes a dative object directly.

So German says:

• jemandem winken

not usually:

• zu jemandem winken for this meaning

That is why the sentence has ihr instead of a prepositional phrase.

This is one of those places where German and English structure the sentence differently:

• English: wave to her
• German: her-DATIVE wave

Because both verbs have the same subject: ich.

German often leaves the subject unstated in the second part when two verbs are joined with und:

• Ich winke ihr zurück und umarme sie später vor dem Haus.

This is more natural than repeating ich.

You could say:

• Ich winke ihr zurück und ich umarme sie später vor dem Haus.

But that sounds more explicit or slightly heavier. The version without the second ich is the normal choice.

Because short pronoun objects often come early in German word order.

So this is the natural order:

• umarme sie später vor dem Haus

A useful rough pattern is:

• verb + pronoun object + time + place

So here:

• umarme = verb
• sie = pronoun object
• später = time
• vor dem Haus = place

If you move sie later, the sentence can sound marked or unnatural unless you are emphasizing something.

Because vor is a two-way preposition.

With two-way prepositions:

• dative = location
• accusative = direction/movement toward somewhere

Here the action happens in front of the house, so it is a location:

• vor dem Haus = dative

Compare:

• Ich stehe vor dem Haus. = location
• Ich gehe vor das Haus. = movement to the area in front of the house

So in your sentence, dem is correct because no movement toward that location is being expressed.

Because German very often uses the present tense for future meaning when the time is already clear from the context.

So später makes it obvious that the hugging happens later, and German does not need a separate future form here.

This is very common:

• Morgen fahre ich nach Berlin.
• Später rufe ich dich an.

So umarme here is grammatically present, but understood as a future action because of später.

zurück adds the idea of back or in return.

So:

• Ich winke ihr. = I wave to her
• Ich winke ihr zurück. = I wave back to her

It usually suggests that she waved first, or that there is some kind of mutual exchange.

Yes, German word order is flexible, but changing it usually changes the focus or emphasis.

For example:

• Später umarme ich sie vor dem Haus.
  This puts more focus on später.

• Ihr winke ich zurück.
  This puts more focus on ihr.

• Vor dem Haus umarme ich sie später.
  This is possible, but sounds more marked and would need the right context.

The original sentence is a very natural, neutral way to say it.

Because the subject is ich, and both verbs are in the 1st person singular present.

The endings are:

• ich winke
• ich umarme

So the sentence matches the subject correctly.

This is especially easy to see if you compare:

• ich winke
• du winkst
• er/sie winkt

and

• ich umarme
• du umarmst
• er/sie umarmt

So both verbs are simply agreeing with ich.