Hieraŭ mia kuzo glitis sur la glacio, sed li ne falis.

In Esperanto, many time words can be used as adverbs without any ending change. Hieraŭ already means yesterday, so it does not need -n.

You may also see the so-called accusative of time with nouns, for example:

• Mi venos lundon. = I will come on Monday.
• Mi restis tie tri tagojn. = I stayed there for three days.

But hieraŭ is already a complete adverb, so hieraŭ is correct as it is.

Yes. Kuzo is the general word for cousin and does not itself specify sex. In this sentence, we know the cousin is male because the next word is li, meaning he.

If you wanted to make it explicitly female, you could say:

• mia kuzino = my female cousin

Traditionally, -ino marks the feminine form.

The ending -is is the Esperanto ending for the past tense.

So:

• gliti = to slip / to slide
• glitis = slipped / was slipping / did slip

and

• fali = to fall
• falis = fell / did fall

Esperanto does not change the verb based on the subject. So:

• mi glitis
• li glitis
• ili glitis

all use the same past-tense ending -is.

They describe two different actions:

• gliti = to slip, slide
• fali = to fall

So the idea is that the cousin slipped on the ice, but did not actually fall down.

This is a very natural distinction in Esperanto, just as in English.

Because sur means on, while en means in.

• sur la glacio = on the ice
• en la glacio = in the ice

If someone slipped while standing on top of an icy surface, sur la glacio is the correct choice.

Because here sur la glacio describes location, not motion toward something.

In Esperanto, the -n ending is often used after a preposition to show direction toward a place:

• Mi iras sur la ponton. = I am going onto the bridge.
• La kato saltis sur la tablon. = The cat jumped onto the table.

But in your sentence, the cousin was already on the ice when he slipped, so it is just location:

• sur la glacio = on the ice

No -n is needed.

La is the definite article, meaning the.

In this sentence, la glacio refers to a specific icy surface relevant to the situation, so the ice sounds natural. Esperanto often uses la much like English does here.

• sur la glacio = on the ice

Without la, sur glacio would sound more general, more like on ice as a substance, rather than on the ice in a specific situation.

Esperanto normally uses an explicit subject pronoun when needed. After sed (but), a new clause begins:

• sed li ne falis = but he did not fall

Unlike some languages, Esperanto does not usually drop subject pronouns just because they are understood from context. So li is included normally and naturally.

Ne means not and usually goes directly before the word it negates. Here it negates the verb:

• li ne falis = he did not fall

This is the standard way to make a sentence negative in Esperanto.

More examples:

• Mi ne scias. = I do not know.
• Ŝi ne venis. = She did not come.

Sed means but. It connects two clauses that contrast with each other:

• mia kuzo glitis sur la glacio
• sed li ne falis

So the contrast is:

• he slipped
• but he did not fall

Its placement is very similar to English but.

The given order is the most natural:

• Hieraŭ mia kuzo glitis sur la glacio, sed li ne falis.

But Esperanto word order is fairly flexible because grammar is shown by endings and function words rather than position alone.

For example, you could also say:

• Mia kuzo hieraŭ glitis sur la glacio, sed li ne falis.

That still makes sense. However, putting hieraŭ first is a common way to emphasize the time: Yesterday...

Esperanto does not usually use a helper verb like English did for ordinary past tense negatives. Instead, it simply uses:

• the past tense ending -is
• plus ne for negation

So:

• falis = fell
• ne falis = did not fall

This is much simpler than English in that respect.

Yes. Gliti can cover the idea of slipping or sliding, depending on context.

In this sentence, because of sur la glacio, an English speaker will usually understand it as slipped on the ice. But the core idea is smooth, uncontrolled movement over a surface.

So the exact English choice can vary, even though the Esperanto sentence is perfectly clear.