Бабушка умеет быстро пеленать малыша, и руки у неё двигаются очень спокойно.

Уметь means to know how to do something / to be skilled at something.
So Бабушка умеет быстро пеленать малыша suggests that grandma has the skill or experience.

Мочь would mean can / be able to, often because circumstances allow it.

• умеет пеленать = she knows how to swaddle
• может пеленать = she is able to swaddle (for some reason, at this moment or in general)

In this sentence, the idea is about experience and competence, so умеет is the natural choice.

After уметь, Russian normally uses the infinitive:

• уметь готовить
• уметь плавать
• уметь пеленать

So умеет пеленать literally means knows how to swaddle.

This is the normal pattern:
уметь + infinitive

Пеленать is the imperfective verb. Here that makes sense because the sentence is talking about a general skill, not one completed action.

• пеленать = to swaddle, as an activity or skill
• запеленать = to swaddle completely / finish swaddling

After уметь, Russian usually prefers the imperfective, because you are describing the ability to do something in general:

• Она умеет пеленать малыша.
• not usually умеет запеленать unless you want to stress successful completion

So the sentence is about grandma’s practiced ability, not one single finished act.

Because малыша is the accusative singular form of малыш, and it is the direct object of пеленать.

Russian masculine animate nouns often have:

• nominative singular: малыш
• accusative singular: малыша

That happens because for masculine animate nouns, the accusative usually looks like the genitive.

Compare:

• Я вижу стол. → стол is inanimate, so accusative = nominative
• Я вижу малыша. → малыш is animate, so accusative = genitive

So here:

• пеленать кого? → малыша

Because руки is the subject of the second clause:

руки у неё двигаются очень спокойно

The hands are the things that are moving, so Russian uses the nominative plural:

• singular nominative: рука
• plural nominative: руки

If you used руками, that would be instrumental plural, meaning with her hands or by means of her hands, which would be a different structure:

• Она двигает руками. = She moves her hands.
• Руки у неё двигаются. = Her hands are moving.

Both are possible, but they are not exactly the same in feel.

• её руки = her hands
• руки у неё = literally something like the hands with her / her hands

Russian often uses у + genitive to express possession, especially in descriptive statements.
So руки у неё двигаются очень спокойно sounds natural and observational, almost like:

• As for her hands, they move very calmly
• Her hands move very calmly

It is especially common when talking about someone’s features, qualities, or body parts.

So:

• её руки двигаются очень спокойно = also correct
• руки у неё двигаются очень спокойно = very natural, slightly more descriptive in tone

This happens because Russian third-person pronouns often gain an initial н- after a preposition.

So:

• её = her
• у неё = at her / of hers / she has

Other examples:

• к нему
• с ней
• у них

This is a standard grammar rule, not a special meaning change.
Here у requires the genitive, and the form becomes неё.

Because двигаться means to move (intransitive), while двигать usually means to move something (transitive).

Compare:

• Руки двигаются. = The hands are moving.
• Она двигает руками. = She moves her hands.

So in your sentence, the hands themselves are the subject, and they are moving, so Russian uses двигаться:

• руки ... двигаются

Also, the ending -ются shows that the subject is plural:

• двигается = singular
• двигаются = plural

Because it describes how the hands move, not what the hands are like.

• спокойно = calmly, smoothly, steadily
• спокойные = calm (adjective, plural)

In the sentence, спокойно modifies the verb двигаются:

• двигаются как? → спокойно

So:

• руки двигаются спокойно = the hands move calmly/smoothly
• руки спокойные = the hands are calm

The second one is grammatically possible, but it sounds strange in this context because hands themselves are not usually described as calm; their movement is.

Russian word order is flexible, but this is a very natural placement.

умеет быстро пеленать малыша puts быстро right next to the action it describes: пеленать.

That is similar to English:

• knows how to swaddle the baby quickly

You could move быстро for emphasis, but the neutral order is very natural as written.

For example:

• Бабушка умеет быстро пеленать малыша. = neutral
• Бабушка быстро умеет пеленать малыша. = possible, but less natural in most contexts
• Бабушка умеет пеленать малыша быстро. = also possible, but the rhythm/emphasis changes

So the sentence uses the most straightforward placement.

Because и is joining two separate clauses, each with its own predicate:

1. Бабушка умеет быстро пеленать малыша
2. руки у неё двигаются очень спокойно

Since these are two full clauses, Russian puts a comma before и.

If и connected only two words or phrases inside one clause, there would usually be no comma. But here it connects two complete parts of the sentence, so the comma is correct.