После ужина я вымыл тёрку и убрал её в шкаф.

Because после requires the genitive case.

• ужин = dinner
• ужина = of dinner / after dinner

So:

• после ужина = after dinner
• после работы = after work
• после фильма = after the film

This is a very common pattern: после + genitive.

Because тёрку is in the accusative case, since it is the direct object of вымыл.

Dictionary form:

• тёрка = a grater

In this sentence:

• я вымыл тёрку = I washed the grater

For many feminine nouns ending in -а, the accusative singular changes to -у:

• книга → книгу
• лампа → лампу
• тёрка → тёрку

So the ending changes because the grater is the thing being washed.

Because её is the accusative (and also genitive) form of the pronoun, while она is the subject form.

• она = she
• её = her / it

Here, её refers to тёрку, which is feminine:

• убрал её в шкаф = put it away in the cupboard

So Russian uses the same form её for her and it when referring to a feminine noun.

Because Russian distinguishes between:

• motion into somewhere → в + accusative
• location in somewhere → в + prepositional

So:

• в шкаф = into the cupboard
• в шкафу = in the cupboard

In this sentence, the grater is being moved from one place to another:

• убрал её в шкаф = put it into the cupboard

Compare:

• Тёрка в шкафу. = The grater is in the cupboard.
• Я убрал тёрку в шкаф. = I put the grater into the cupboard.

Because these are perfective verbs, which emphasize a completed action.

• мыть = to wash (imperfective)

• вымыть = to wash completely / wash off (perfective)

• убирать = to put away / tidy up (imperfective)
• убрать = to put away / remove / tidy up completely (perfective)

In this sentence, the speaker is describing completed actions:

1. He washed the grater.
2. He put it away in the cupboard.

So perfective verbs are natural here:

• вымыл
• убрал

If you used imperfective forms, the meaning would shift more toward process, repetition, or background description.

Here it means put away.

The verb убрать can have several related meanings, such as:

• remove
• put away
• tidy up
• clean up

But the phrase убрал её в шкаф makes the meaning clear: he put it away into the cupboard.

So the context tells you which English translation fits best.

Because both actions have the same subject: я.

Russian often uses one subject with multiple verbs connected by и:

• Я пришёл и сел. = I came and sat down.
• Я открыл окно и вышел. = I opened the window and went out.

So:

• я вымыл тёрку и убрал её в шкаф

means:

• I washed the grater and put it away in the cupboard

Repeating я would usually be unnecessary.

Yes. As written, it suggests the speaker is male.

In the Russian past tense, verbs agree in gender in the singular:

• masculine: вымыл, убрал
• feminine: вымыла, убрала
• neuter: вымыло, убрало

So:

• После ужина я вымыл тёрку и убрал её в шкаф.
  = spoken by a man

If the speaker were a woman, it would be:

• После ужина я вымыла тёрку и убрала её в шкаф.

Тёрка means grater — the kitchen tool used for grating things like cheese, carrots, or potatoes.

It comes from the verb тереть = to rub / grate.

A useful connection is:

• тереть сыр = to grate cheese
• тёрка = grater

Also note the spelling and stress:

• тёрка
• the ё is stressed and pronounced like yo

Yes. Russian word order is more flexible than English word order because case endings show the grammatical roles.

The sentence could also be said as:

• Я после ужина вымыл тёрку и убрал её в шкаф.
• Тёрку я вымыл после ужина и убрал её в шкаф.

These versions are still grammatical, but the emphasis changes slightly.

The original sentence,

• После ужина я вымыл тёрку и убрал её в шкаф, is very natural and starts with the time expression after dinner.

Usually it means after dinner in the general sense, just like in English.

Russian often does not use articles, so context determines whether something is specific or general. In most everyday situations:

• после ужина = after dinner

If the context were very specific, English might translate it as after the dinner, but in ordinary use after dinner is the best translation.

Sometimes Russian can omit an object if it is very obvious from context, but in this sentence её is very natural and helps the sentence sound complete and clear.

• убрал её в шкаф = put it away in the cupboard

Without её, the sentence would sound less standard here, because убрать в шкаф normally wants an object: put what into the cupboard?

So keeping её is the normal choice.

A few things:

• тёрку has ё, so it is stressed: ТЁР-ку
• её is pronounced roughly yi-YO
• ужина is stressed У-жи-на
• вымыл is stressed on the first syllable: ВЫ-мыл
• убрал is stressed on the last syllable: у-БРАЛ

Also, in normal speech, short unstressed words like я, и, в are pronounced very lightly and attach closely to neighboring words.

So the rhythm is something like:

После УЖИНА я ВЫмыл ТЁРку и уБРАЛ её в шкаф.