Электросчётчик и показания счётчика уже проверены.

Because the subject is a combination of two things joined by и:

• Электросчётчик = the electric meter
• показания счётчика = the meter readings

Together they form a plural subject, so the predicate is plural too:

• проверен = checked, masculine singular
• проверена = checked, feminine singular
• проверено = checked, neuter singular
• проверены = checked, plural

So:

• Электросчётчик уже проверен. = The meter has already been checked.
• Показания счётчика уже проверены. = The meter readings have already been checked.
• Электросчётчик и показания счётчика уже проверены. = The meter and the meter readings have already been checked.

The main nouns электросчётчик and показания are in the nominative case, because they are the subject of the sentence.

Breakdown:

• электросчётчик — nominative singular
• показания — nominative plural
• счётчика — genitive singular

So the structure is:

• Электросчётчик = subject
• показания счётчика = subject
• уже проверены = predicate

Because показания usually takes a dependent noun in the genitive, meaning readings of something.

So:

• показания счётчика = readings of the meter
• показания температуры = temperature readings
• показания прибора = instrument readings

This is a very common Russian pattern: a noun followed by another noun in the genitive to show possession or relation.

In this context, Russian normally uses показания in the plural to mean readings.

This is the standard expression for things like utility meter readings:

• передать показания счётчика = to submit meter readings
• снять показания = to take a reading / record the readings

Even if English sometimes says the meter reading in the singular, Russian very often prefers the plural показания, because it is thought of as a set of numerical values shown by the device.

The singular показание exists, but in everyday usage it often means indication or even testimony in legal contexts, so for utility meters the plural is much more natural.

Счётчик is the general word for a meter, counter, or measuring device.

Электросчётчик is more specific: it means an electricity meter.

In this sentence, the first mention uses the full, precise word:

• Электросчётчик = the electricity meter

Then the second mention uses the shorter счётчика in the phrase показания счётчика, because once the context is clear, Russian often uses the shorter noun naturally.

So this is normal and not inconsistent.

Russian usually omits the present-tense form of to be.

So a sentence like:

• ... уже проверены

literally looks like already checked, but in natural English it often means:

• are already checked
• have already been checked

depending on context.

Russian is focusing on the resulting state: the checking is complete.

Compare:

• Электросчётчик уже проверен. = The meter is already checked / has already been checked.
• Электросчётчик был проверен. = The meter was checked / had been checked.
• Электросчётчик будет проверен. = The meter will be checked.

So in the present, Russian often leaves to be unspoken.

Проверены is a short-form passive participle from the verb проверить (to check, perfective).

It is used predicatively, meaning it acts as the main predicate of the sentence.

A useful contrast:

• проверенные документы = checked documents
  
  • long form, used like an adjective before a noun
• документы проверены = the documents have been checked
  
  • short form, used as the predicate

So in your sentence:

• Электросчётчик и показания счётчика уже проверены.

the short form tells us the action has been completed and the result now holds.

Уже means already, and here it modifies the predicate проверены.

So:

• уже проверены = already checked

This is a very natural position in Russian. The sentence has a neutral word order:

• Электросчётчик и показания счётчика уже проверены.

Russian word order is flexible, though, so other orders are possible if the speaker wants a different emphasis:

• Уже проверены электросчётчик и показания счётчика.
• Электросчётчик и показания счётчика проверены уже.

These are possible, but the original version is the most neutral and standard.

The stressed vowel in счётчик is pronounced ё, so the pronunciation is roughly shchyotchik.

In printed Russian, ё is often replaced by е, so you may also see:

• электросчетчик
• счетчика

But the pronunciation still remains ё.

For learners, it is helpful that this sentence writes ё explicitly, because it shows the correct pronunciation and avoids confusion.

Yes. That is exactly how the grammar is built.

The coordinated subject is:

• Электросчётчик
• и
• показания счётчика

So Russian presents them as two separate items that have both undergone checking.

This does not necessarily mean two completely different procedures in real life, but grammatically it means both the meter itself and the meter readings are included in what has been checked.