На катке я замёрз и решил поехать домой.

Russian uses different prepositions depending on how a place is conceptualized.

With каток (ice rink), Russian normally uses на, not в:

• на катке = at the ice rink / on the rink

This is similar to how Russian says:

• на стадионе = at the stadium
• на концерте = at the concert
• на работе = at work

So this is something you largely learn as a standard collocation: быть на катке.

На катке uses the prepositional case.

The noun is:

• каток = ice rink

After на when it means at/on/in a place, Russian usually takes the prepositional:

• каток → на катке

This is why the ending changes from -ок to -ке.

So:

• каток = nominative
• на катке = prepositional, meaning at the rink

Замёрз is the past tense of замёрзнуть, which means to get cold, to become cold, or to freeze.

In this sentence, it means something like:

• I got cold
• I became too cold

It focuses on the result: the speaker reached the state of being cold.

That fits the context well:

• At the ice rink, I got cold and decided to go home.

This is a very common aspect question.

• мёрз comes from мёрзнуть and emphasizes the process/state: I was feeling cold / I was freezing
• замёрз comes from замёрзнуть and emphasizes the result: I got cold / ended up cold

Compare:

• Я мёрз на катке. = I was cold / freezing at the rink.
• Я замёрз на катке. = I got cold at the rink.

In your sentence, замёрз makes sense because it leads to a consequence:

• I got cold, so I decided to go home.

Both are possible in some contexts, but решил поехать домой is very natural here.

• решил = decided
• поехать = to set off/go by transport

The infinitive after решил is often chosen based on aspect and nuance.

поехать

This is a perfective verb of motion. It often means:

• to set off
• to go
• to leave by vehicle

So решил поехать домой means:

• decided to go home
• more literally, decided to head home

ехать

This is the imperfective partner, focusing more on the process of traveling.

So:

• решил поехать домой = decided to leave/head home
• решил ехать домой = decided that he would be going home / chose to travel home

In this sentence, поехать sounds more natural because it expresses the decision to begin the trip.

Yes, usually.

Russian distinguishes between different kinds of motion verbs:

• идти / пойти = to go on foot
• ехать / поехать = to go by transport

So поехать домой suggests going home by:

• car
• bus
• metro
• taxi
• train
• maybe even skates or some vehicle-like means, depending on context

If the speaker were clearly walking home, Russian would more likely use:

• решил пойти домой

Домой is an adverb meaning homeward / to home.

Russian often uses special forms for home:

• дома = at home
• домой = home, toward home

So:

• Я дома. = I am at home.
• Я иду домой. = I am going home.

That is why there is no preposition like to. English says go home, and Russian does something similar with домой.

Yes, but the meaning would change slightly.

• пойти домой = to go home on foot / to set off home
• поехать домой = to go home by transport

So the choice depends on how the person is leaving.

Examples:

• Я замёрз и решил пойти домой. = I got cold and decided to walk home.
• Я замёрз и решил поехать домой. = I got cold and decided to go home by transport.

In Russian past tense, singular verbs agree with the gender of the subject.

Here the speaker is presumably male, so you get:

• я замёрз
• я решил

If the speaker were female, it would be:

• я замёрзла
• я решила

So the sentence would become:

• На катке я замёрзла и решила поехать домой.

Yes, very often.

Russian frequently omits personal pronouns when the subject is clear from the verb form or context.

So you could say:

• На катке замёрз и решил поехать домой.

This can sound natural in the right context, especially in conversation or narration.

However, я is still perfectly correct. It may be included for:

• clarity
• emphasis
• contrast

Yes, Russian word order is flexible.

The sentence starts with На катке to set the scene first:

• At the rink, I got cold and decided to go home.

Other possible orders include:

• Я замёрз на катке и решил поехать домой.
• Я на катке замёрз и решил поехать домой.

These all mean roughly the same thing, but the focus shifts a little.

• На катке... puts the location first.
• Я... puts the speaker first.

Russian often uses word order to manage emphasis rather than basic grammar.

И simply means and.

It joins two past-tense actions:

• замёрз = got cold
• решил поехать домой = decided to go home

So the structure is:

• [At the rink] [I got cold] and [decided to go home].

The two actions are connected in a natural sequence:

1. I got cold.
2. As a result, I decided to go home.

The standard form is замёрз with ё.

In real Russian writing, especially informal or printed text, ё is often written as е, so you may also see:

• замерз

But the pronunciation and meaning are still those of замёрз.

For learners, it is helpful to remember the correct form with ё, because it tells you the stress and pronunciation more clearly:

• за-МЁРЗ

So yes, native texts may omit the dots, but the underlying word is still замёрз.