После бега мне хотелось не на каток, а домой, потому что одно ухо всё ещё мёрзло.

Because после requires the genitive case.

• бег = running / a run
• после бега = after the run / after running

So бег changes to бега because it is in the genitive singular.

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

Мне хотелось is a very natural Russian way to say I felt like..., I wanted to..., or I had a desire to...

Literally, it is structured more like:

• to me it was wanted
• or more naturally: I felt like

So:

• я хотел = I wanted
• мне хотелось = I felt like / I wanted

The second version often sounds a bit softer, more like a passing feeling or urge.

In this sentence, мне хотелось fits well because the speaker is describing how they felt after running.

Because this is an impersonal construction.

In мне хотелось, there is no normal subject like я. The verb is used impersonally, and in the past tense such verbs often appear in the neuter singular form:

• мне хотелось = I felt like
• ему не спалось = he couldn’t sleep
• ей не сиделось = she felt restless / she couldn’t sit still

So хотелось is not agreeing with мне. It is just the standard neuter past form used in this kind of impersonal sentence.

Yes, you can understand an omitted verb of motion, usually something like идти or поехать, depending on context.

So:

• мне хотелось не на каток, а домой

really means something like:

• мне хотелось идти не на каток, а домой
• I felt like going not to the skating rink, but home

Russian often omits an infinitive when the destination alone makes the meaning obvious.

This is the standard Russian pattern for not X, but Y:

• не ..., а ...

So:

• не на каток, а домой = not to the skating rink, but home

The conjunction а is used here to show contrast.

Compare:

• не чай, а кофе = not tea, but coffee
• не сегодня, а завтра = not today, but tomorrow

Because каток here means skating rink, and Russian normally uses на with places seen as venues, surfaces, or events of this type.

• на каток = to the skating rink
• на стадион = to the stadium
• на концерт = to the concert

Since this expresses motion toward the place, каток is in the accusative case. For an inanimate masculine noun like каток, the accusative looks the same as the nominative, so it stays каток.

Домой is a very common adverb meaning homeward / home.

It is used when someone is going home:

• идти домой = to go home
• ехать домой = to go home
• хотелось домой = I felt like going home

Compare:

• дома = at home
• домой = home, toward home
• в дом = into the house
• к дому = toward the house

So домой is the natural choice if the meaning is simply home as a destination.

Adding одно means one ear, which sounds a little more vivid and specific.

• ухо мёрзло = an ear was cold / the ear was freezing
• одно ухо мёрзло = one ear was still freezing

Russian often uses one in places where English might or might not use it. Here it emphasizes that specifically one ear was still cold.

Also, ухо is a neuter noun, so one must also be neuter:

• masculine: один
• feminine: одна
• neuter: одно

So:

• одно ухо

Because мёрзло agrees with ухо, which is neuter singular.

The verb мёрзнуть means to be cold, to freeze, or to get cold.

Past tense forms:

• мёрз — masculine
• мёрзла — feminine
• мёрзло — neuter
• мёрзли — plural

Since ухо is neuter, we get:

• ухо мёрзло = the ear was freezing / the ear was still cold

Also, you may sometimes see мерзло without ё in writing. Russian often omits the dots over ё, but the pronunciation is still мёрзло.

Всё ещё means still.

So:

• одно ухо всё ещё мёрзло = one ear was still freezing / one ear was still cold

It shows that the condition continued from earlier.

Other examples:

• Я всё ещё жду. = I’m still waiting.
• Он всё ещё спит. = He’s still sleeping.

Russian word order is flexible, and here the order helps the sentence sound natural and expressive.

• После бега sets the time/context first: After the run
• мне хотелось introduces the feeling: I felt like
• не на каток, а домой gives the contrast: not to the skating rink, but home
• потому что... adds the reason: because...

So the sentence flows as:

1. context
2. feeling
3. contrast
4. reason

That is very typical Russian sentence organization.

Yes, it could, but the tone would be a bit different.

• Я хотел домой = I wanted to go home
• Мне хотелось домой = I felt like going home / I wanted to go home

Both are correct. But мне хотелось often sounds a bit more about an internal feeling or urge, and a bit less direct or categorical than я хотел.

In this sentence, that softer feeling works well because the speaker is explaining how they felt after running.

It can mean either, depending on context.

• after running
• after the run

Russian often uses a noun where English might use either a noun phrase or a gerund. So бег can refer to the activity of running or to a specific run.

In this sentence, both interpretations fit well:

• after running
• after the run

The exact nuance comes from the wider context.