Gölette üç ördek yüzüyor, kıyıda da iki civciv annelerini bekliyor.

They are both nouns with the locative ending, which usually means in, at, on.

• gölet = pond

• gölette = in the pond

• kıyı = shore, edge
• kıyıda = on/at the shore

The locative ending changes shape because of vowel harmony and consonant rules:

• -de / -da
• -te / -ta

So gölet ends in voiceless t, which is why it becomes gölet-te.
kıyı ends in a vowel, so it takes -da.

Because these are two different things:

• attached -da / -de / -ta / -te = the locative ending, meaning in / at / on
• separate da / de = also, too, as well

So:

• kıyıda = on the shore
• kıyıda da = also on the shore or and on the shore

A very common learner mistake is to confuse these. The locative is written attached, but the also word is written separately.

In Turkish, a noun after a number is normally singular in form.

So you say:

• üç ördek = three ducks
• iki civciv = two chicks

Not:

• üç ördekler
• iki civcivler

The number already shows that there is more than one, so Turkish usually does not add the plural ending -lar / -ler as well.

They are in the present continuous or progressive tense, formed with -(I)yor.

• yüzmek → yüzüyor = is swimming / are swimming
• beklemek → bekliyor = is waiting / are waiting

A couple of useful points:

• Turkish uses the same basic verb form for he/she/it and they unless a plural ending is added.
• In bekliyor, the stem-final vowel drops:
  • bekle-
    • -yor → bekliyor

Because with subjects that contain a number, Turkish normally uses a singular verb.

So these are standard:

• üç ördek yüzüyor
• iki civciv bekliyor

The number already tells you the subject is plural, so the verb usually stays singular.

This is very natural Turkish, even though it feels unusual from an English point of view.

Because the possessor is plural: iki civciv = two chicks.

Compare:

• annesi = his/her mother
• anneleri = their mother or sometimes his/her mothers, depending on context

Here the meaning is clearly their mother, so Turkish uses anneleri.

Then with the accusative ending added, it becomes annelerini.

Even if the two chicks share just one mother, Turkish still uses the plural possessor form because the possessors are plural.

That final part marks the noun as a definite direct object.

Breakdown:

• anne = mother
• anneleri = their mother
• annelerini = their mother, as the object of the verb

Here, the chicks are waiting for a specific person, so Turkish marks that object.

Also, after a possessive ending, Turkish often uses a buffer n before a case ending:

• annesi → annesini
• anneleri → annelerini

So the -ni here is not a separate word meaning their. It is part of the object marking.

Yes. Turkish word order is flexible.

The original sentence starts with the locations:

• Gölette üç ördek yüzüyor
• kıyıda da iki civciv annelerini bekliyor

This gives a nice scene-setting feel, almost like describing a picture.

A different but still natural version could be:

• Üç ördek gölette yüzüyor, iki civciv de kıyıda annelerini bekliyor.

That version puts the subjects earlier. The meaning stays basically the same, but the emphasis shifts a little.