Я посчитал яблоки в корзине, и их оказалось десять.

Посчитал is the perfective form, so it presents the counting as a completed action with a result.

Here the speaker finished counting and learned the number. That fits the second clause: их оказалось десять.

By contrast, считал would usually suggest:

• the process of counting,
• a repeated/habitual action,
• or an unfinished background action.

Also, посчитать can sometimes mean to consider / to think, but in this sentence, with яблоки as the object, it clearly means to count.

That is the past tense masculine singular form.

With я, Russian past tense shows the speaker’s gender:

• я посчитал = a male speaker
• я посчитала = a female speaker

So the sentence, as written, sounds as if the speaker is male.

Because яблоки is the direct object of посчитал.

The noun яблоко is inanimate, and in Russian the accusative plural of inanimate nouns is the same as the nominative plural:

• nominative plural: яблоки
• accusative plural: яблоки

So посчитал яблоки is exactly what you should expect.

Яблок is genitive plural, and you would use it in other patterns, for example:

• десять яблок
• нет яблок
• много яблок

Because this phrase shows location, not direction.

• в корзине = in the basket / inside the basket
• в корзину = into the basket

Here the apples are already located there, so Russian uses в + prepositional:

• в корзине

If there were motion into the basket, then you would use в + accusative:

• положить яблоки в корзину

Here оказалось comes from оказаться, which often means:

• to turn out
• to prove to be
• to be found to be

So их оказалось десять means something like:

• there turned out to be ten of them
• they turned out to number ten

It does not mean that something physically appeared. It is about the result that was discovered.

Russian often uses a pronoun here to avoid repeating the noun.

So:

• Я посчитал яблоки в корзине, и их оказалось десять.

means:

• I counted the apples in the basket, and there turned out to be ten of them.

Using их sounds natural because the apples were already mentioned.

You could also say:

• Я посчитал яблоки в корзине, и яблок оказалось десять.

That is also grammatical, but the version with их is smooth and idiomatic.

Because this pattern works like ten of them, not they were ten.

In их оказалось десять, их is best understood as of them in a quantitative construction:

• их оказалось десять = there turned out to be ten of them

If you used они, that would suggest a normal personal subject, but that is not how this sentence is built.

This is similar to other Russian patterns such as:

• нас было трое = there were three of us
• их пришло двое = two of them came

Because this is an impersonal quantitative construction.

Russian often uses neuter singular with verbs like было, стало, оказалось when the sentence is about the resulting number/quantity:

• их оказалось десять
• нас было пятеро
• гостей пришло много

So the idea is not really the apples turned out... with a normal plural subject. It is more like:

• the number turned out to be ten
• there turned out to be ten of them

That is why оказалось is neuter singular.

Because in this kind of sentence the numeral is normally in the form used for the quantity itself:

• их оказалось десять
• нас было трое
• яблок оказалось пять

Here десять is the quantity that was discovered.

Десяти would belong to a different grammatical environment, for example after a preposition or in certain governed-case patterns, but not in this basic there turned out to be ten structure.

Because the sentence contains two separate clauses, each with its own finite verb:

1. Я посчитал яблоки в корзине
2. их оказалось десять

In Russian, when и joins two independent clauses like that, a comma is normally used.

So the comma here is standard punctuation.

Yes. Russian word order is fairly flexible, and changing it can shift the focus.

The given sentence is natural because it first gives:

1. the action: I counted
2. the thing counted: the apples in the basket
3. the discovered result: there were ten

Other natural variants are possible, for example:

• В корзине оказалось десять яблок.
• Я посчитал яблоки в корзине: их оказалось десять.

But the original version is a very normal way to tell the story step by step.