Ég klæði mig fyrst og klæði síðan dóttur mína.

Because sig is the reflexive form used for the 3rd person only, while the actual reflexive/object pronoun changes with the subject.

So:

• Ég klæði mig = I get dressed / I dress myself
• Þú klæðir þig = you get dressed
• Hann / hún klæðir sig = he / she gets dressed

So the dictionary gives að klæða sig as the general reflexive pattern, but with ég you must use mig, not sig.

Klæði is the 1st person singular present tense form of að klæða.

So:

• ég klæði = I dress / I get dressed
• þú klæðir = you dress
• hann / hún klæðir = he / she dresses

In this sentence, both instances of klæði mean I dress; the difference is in the object:

• Ég klæði mig = I dress myself / I get dressed
• (Ég) klæði dóttur mína = I dress my daughter

Because ég is the subject form, but mig is the object form.

Compare:

• Ég klæði mig = I dress myself
• Hún sér mig = She sees me

In Ég klæði mig, ég is the person doing the action, and mig is the object of the verb. English also distinguishes I and me, so this is a similar idea.

Because the subject is still the same: I.

Icelandic, like English, often leaves out the repeated subject in coordinated clauses when it is obvious:

• Ég klæði mig fyrst og klæði síðan dóttur mína.

This works like English I get dressed first and then dress my daughter.
You could repeat the subject, but it is not necessary:

• Ég klæði mig fyrst og ég klæði síðan dóttur mína.

That version is grammatical, but less natural here because the repeated ég is unnecessary.

Because Icelandic usually keeps the finite verb in an early position, especially in main clauses.

Here the second clause is shortened because the subject ég is omitted:

• og klæði síðan dóttur mína

That is a natural way to say and then dress my daughter.

If you include síðan earlier, you would normally also include the subject:

• og síðan klæði ég dóttur mína

So the sentence you were given is a compact, natural coordinated structure.

Because dóttur mína is the accusative form, and this phrase is the direct object of klæði.

The basic dictionary form is:

• dóttir mín = my daughter

But after a verb like að klæða, the direct object goes in the accusative:

• Ég klæði dóttur mína = I dress my daughter

So both words change as needed:

• dóttir → dóttur
• mín → mína

Because noun + possessive pronoun is a very common and natural Icelandic pattern.

So Icelandic often says:

• dóttir mín = my daughter
• maðurinn minn = my husband
• húsið mitt = my house

In your sentence, that same pattern appears in the accusative:

• dóttur mína

Placing the possessive before the noun is much less neutral and often sounds contrastive or emphatic. For an ordinary statement, dóttur mína is the natural choice.

Because the possessive already makes the noun definite.

In English, my daughter is already definite, and Icelandic works similarly. So you do not need a separate word meaning the here.

That is why Icelandic says:

• dóttir mín = my daughter, not the my daughter

The possessive phrase itself is enough.

They show the order of actions:

• fyrst = first
• síðan = then / afterwards

So the sentence is structured as:

1. Ég klæði mig fyrst = First I get dressed
2. og klæði síðan dóttur mína = and then I dress my daughter

They are very common time-ordering adverbs in Icelandic.

It can mean both, but in everyday English the most natural translation is often I get dressed.

Literally, it is I dress myself, but Icelandic commonly uses this reflexive structure where English prefers get dressed.

So:

• Ég klæði mig á morgnana
  = I get dressed in the mornings

A very literal translation is possible, but get dressed is usually the more natural English equivalent.

Because repeating it makes the structure clear and natural in Icelandic:

• Ég klæði mig fyrst og klæði síðan dóttur mína.

This clearly shows two actions:

1. I dress myself
2. I dress my daughter

Since the first object is reflexive (mig) and the second is a different direct object (dóttur mína), repeating the verb helps avoid ambiguity. It is the safest and most standard way to say it.