После колыбельной дочка быстро засыпает, а днём уже ползает по ковру и не даёт себя долго укачивать.

Because после requires the genitive case.

• Dictionary form: колыбельная
• Genitive singular: колыбельной

So:

• после колыбельной = after the lullaby

This is a very common pattern:

• после урока = after the lesson
• после обеда = after lunch
• после работы = after work

Even though колыбельная looks like an adjective, here it is being used as a noun meaning lullaby.

Дочка is an affectionate, everyday form of дочь.

• дочь = daughter
• дочка = daughter, little daughter, dear daughter

In many contexts, Russian prefers these more natural, affectionate family words:

• мама instead of мать
• папа instead of отец
• дочка instead of дочь
• сынок or сын depending on tone

So дочка sounds warm and natural here, especially in a sentence about a small child.

Russian often uses the present tense to describe habitual, typical, or repeated actions.

So this sentence is not necessarily about one specific moment. It describes what usually happens:

• after a lullaby, the daughter falls asleep quickly
• during the day, she crawls around on the carpet
• and she doesn’t let herself be rocked for long

This is similar to English sentences like:

• After dinner, he usually gets sleepy.
• In the morning, she drinks coffee.

So the present tense here has a general/habitual meaning.

They are different actions:

• засыпает = falls asleep, is falling asleep
• спит = is asleep, sleeps

So быстро засыпает means she enters sleep quickly. It does not mean simply that she sleeps quickly.

This is a very important distinction in Russian:

• уснуть / засыпать = to fall asleep
• спать = to sleep

Днём is the instrumental form of день, and in this use it functions as an adverb meaning:

• in the daytime
• during the day

This is a fixed and very common time expression:

• днём = by day / during the day
• ночью = at night
• утром = in the morning
• вечером = in the evening

So you do not need to interpret it literally every time as a normal instrumental noun. It is best learned as a set time expression.

This is about the difference between a repeated/habitual action and a single ongoing action.

• ползает = crawls around, crawls habitually, is able to crawl about
• ползёт = is crawling right now in one direction / one instance

Since the sentence describes what the child typically does during the day, ползает fits better.

Compare:

• Ребёнок уже ползает. = The baby already crawls / has started crawling.
• Ребёнок ползёт к двери. = The baby is crawling toward the door.

Because по often takes the dative when it means movement over, along, or around on a surface or area.

• ковёр = carpet
• Dative singular: ковру

So:

• ползает по ковру = crawls around on the carpet

This is a very common pattern:

• ходить по улице = walk along the street
• бегать по парку = run around the park
• ползти по полу = crawl along the floor

Here the idea is not movement to a destination, but movement within/over an area.

Уже means already.

Here it suggests a development or contrast:

• after the lullaby she falls asleep quickly,
• but during the day she is already crawling around on the carpet

With children, уже often highlights progress, a new stage, or an unexpectedly advanced ability.

For example:

• Он уже ходит. = He already walks.
• Она уже говорит. = She already talks.

So in this sentence, уже makes the daytime behavior sound a bit more vivid: she is no longer just lying there; she is already active and mobile.

This is a very useful Russian pattern:

• давать / не давать + object + infinitive

It means:

• to let / allow
• to not let / not allow

So:

• не даёт ... укачивать = doesn’t let ... be rocked

The full structure is:

• не даёт себя долго укачивать

Literally:

• does not allow herself to be rocked for long

More naturally in English:

• she doesn’t let people rock her for long
• she won’t be rocked for long

This construction can feel unusual to English speakers, but it is very common in Russian.

Compare:

• Он не даёт себя фотографировать. = He won’t let people photograph him.
• Она не даёт себя перебивать. = She doesn’t let people interrupt her.

Себя is the reflexive pronoun, meaning something like oneself / herself / himself, depending on the subject.

Here the subject is дочка, so себя refers back to her:

• дочка ... не даёт себя укачивать
  = the daughter does not let herself be rocked
  = more naturally, she does not let others rock her

Why is it needed? Because укачивать needs an object: someone is rocking her. Russian uses себя to show that the object is the same person as the subject of the main verb.

So the logic is:

• she does not allow people to rock her

Because the sentence is talking about an action as a process, not a completed result.

• укачивать = imperfective
• укачать = perfective

With долго, the focus is on duration:

• не даёт себя долго укачивать = she doesn’t let people rock her for long

The imperfective fits well with:

• repeated attempts
• ongoing action
• duration

If you used the perfective here, it would sound more like a completed result, which is not the point.

А often links two parts with a sense of contrast, switch of focus, or meanwhile, while и is usually a simpler and.

Here the sentence contrasts two situations:

• after the lullaby she quickly falls asleep,
• but / whereas / meanwhile during the day she is already crawling around and resisting being rocked

So а is very natural because it marks a change in scene or behavior.

Very roughly:

• и = and
• а = and/but/whereas, with contrast or shift

Russian often omits words that are obvious from context.

In this sentence, дочка is the subject of all the verbs:

• дочка быстро засыпает
• (дочка) ползает
• (дочка) не даёт себя укачивать

Repeating дочка again would be unnecessary unless the speaker wanted extra emphasis.

This kind of omission is very normal in Russian and often sounds more natural than repeating the noun every time.

The basic meaning stays the same, but the word order helps shape emphasis and rhythm.

• дочка быстро засыпает sounds neutral: quickly is part of the main statement.
• не даёт себя долго укачивать puts долго right next to укачивать, which naturally emphasizes the length of the rocking.

Russian word order is flexible, but not random. It often reflects:

• what is already known
• what is being emphasized
• what sounds most natural in context

So the order here is quite natural and smooth for spoken or written Russian.