Ребёнок любит бегать по дому босиком, и мама всё время просит его надеть тапки.

Because бегать and бежать are different kinds of to run.

• бегать = to run around / to run habitually / to run repeatedly
• бежать = to be running in one specific direction, at one specific moment

Here the idea is that the child likes running around the house in general, so любит бегать is the natural choice.

По дому means around the house or through the house, with a sense of movement from place to place.

• по дому = around the house
• в доме = in the house

Since the child is running around, Russian uses по + dative very naturally here.

Because the preposition по often takes the dative case when it means movement around/along/through an area.

So:

• дом = nominative
• дому = dative

That is why you get по дому.

Босиком means barefoot.

In Russian, this is an adverb-like word, so you can say:

• ходить босиком = to walk barefoot
• бегать босиком = to run barefoot

English often uses an adjective after the verb, but Russian commonly uses this fixed form босиком without any preposition.

Because this sentence joins two full clauses:

• Ребёнок любит бегать по дому босиком
• мама всё время просит его надеть тапки

Each part has its own subject and predicate, so Russian normally puts a comma before и, just as English often does before and when joining two independent clauses.

Всё время means all the time, constantly, or keeps doing something.

So:

• мама всё время просит = Mom is always asking / keeps asking

It does not literally focus on clock time here; it expresses repetition and annoyance or persistence.

Его is in the accusative case here, because просить usually takes the person being asked in the accusative:

• просить кого? = to ask whom?

So:

• мама просит его = Mom asks him

Then the infinitive explains what she asks him to do:

• просит его надеть тапки = asks him to put on slippers

This is a very common learner question.

• надеть = to put on something on yourself or on someone
• одеть = to dress someone

So you say:

• надеть тапки = to put on slippers
• одеть ребёнка = to dress the child

A useful memory trick is:

• надеть что-то
• одеть кого-то

Here тапки are the thing being put on, so надеть is correct.

Because the mother is asking for a single completed action: put the slippers on.

• надевать = imperfective, process or repeated action
• надеть = perfective, one completed act

After просить + infinitive, both aspects are possible depending on meaning, but here the natural meaning is: Please put your slippers on. That points to надеть.

Тапки is the accusative plural, and for inanimate nouns it looks the same as the nominative plural.

• nominative plural: тапки
• accusative plural: тапки

Since надеть что? takes the accusative, тапки is correct.

The singular would be тапок only in some colloquial or counting contexts, but normally slippers are talked about as a pair, so Russian often uses the plural.

They are very close, but there is a slight difference in tone.

• тапки = slippers
• тапочки = slippers, but more affectionate, cozy, or diminutive

A mother might easily say either one. Тапочки can sound softer or more domestic.

Russian often states the subject explicitly when introducing a new clause, especially to make the contrast clear:

• the child likes doing one thing
• the mother keeps asking for the opposite

So и мама clearly sets up the second participant. Leaving it out would be less clear and less natural in this sentence.

Yes. Russian word order is flexible, though the original sounds natural and neutral.

For example:

• Ребёнок любит босиком бегать по дому...
• Мама всё время просит его надеть тапки.
• Всё время мама просит его надеть тапки...

But word order changes emphasis. The original version is a straightforward, neutral way to say it.

The correct pronunciation is with ё: ребёнок.

In many Russian texts, ё is often written as е, so you may see ребенок. However, learners should remember that the sound is still ё.

The same applies to всё, which is often written все in ordinary typing, though that can create ambiguity.