Терморегулятор и термоголовка уже проверены.

Because the subject is two things joined by и: терморегулятор и термоголовка.

In Russian, when two singular nouns are connected by и and act together as the subject, the predicate normally goes in the plural:

• стол и стул готовы — the table and chair are ready
• мама и папа дома — mom and dad are at home

So:

• терморегулятор
  • термоголовка = plural subject
• therefore проверены = plural

In the plural, Russian does not distinguish masculine vs. feminine here, so one plural form works for both.

Проверены is the short-form passive participle, plural, from проверить (to check / inspect / verify).

It means something like:

• are checked
• have been checked
• stand checked

In this sentence, it describes the state resulting from a completed action: the thermostat and the thermostatic head have already been checked.

A useful way to think of it:

• проверить — to check
• проверенный — checked, verified (full form)
• проверены — are/have been checked (short predicate form, plural)

Because Russian usually uses the short form when the participle is the main predicate of the sentence.

Here проверены is the predicate:

• Терморегулятор и термоголовка уже проверены.

The full form проверенные is normally used like an adjective, especially before a noun:

• проверенные детали — checked parts
• проверенные устройства — verified devices

So a simple rule is:

• short form = used predicatively: они проверены
• full form = used attributively: проверенные устройства

Because Russian normally omits the present-tense form of to be.

English needs:

• are checked
• have been checked

Russian often does not:

• они проверены

So Терморегулятор и термоголовка уже проверены is a complete sentence even without a separate word for are.

If you want to make the past more explicit, Russian can add были:

• Терморегулятор и термоголовка уже были проверены.

That sounds more clearly like were / had been checked as a past event.

They are in the nominative case because they are the subject of the sentence.

You can tell from their basic dictionary forms:

• терморегулятор — nominative singular
• термоголовка — nominative singular

Together they form a compound subject:

• терморегулятор и термоголовка

Since they are the things being talked about, nominative is exactly what you would expect.

Their endings give strong clues:

• терморегулятор ends in a consonant, so it is normally masculine
• термоголовка ends in -а, so it is normally feminine

That is why, if each word stood alone, you would get:

• Терморегулятор уже проверен. — masculine singular
• Термоголовка уже проверена. — feminine singular

But together they become plural:

• Терморегулятор и термоголовка уже проверены.

Because this sentence focuses on the items, not on who checked them.

Russian often uses a passive structure when the agent is unknown, unimportant, or obvious from context.

So:

• Терморегулятор и термоголовка уже проверены. = The thermostat and the thermostatic head have already been checked.

If you say проверили, then you need an active idea like someone checked them:

• Терморегулятор и термоголовку уже проверили.

Notice something important here: in the active version, the objects would no longer stay in the nominative:

• терморегулятор → терморегулятор
• термоголовка → термоголовку

So the sentence structure changes quite a bit.

Уже means already.

In this sentence it tells you that the checking has been completed before the relevant moment:

• уже проверены — already checked

Its position is quite natural here, right before the predicate:

• Терморегулятор и термоголовка уже проверены.

You can move it around for emphasis, but the neutral version is the one you have here.

For example:

• Уже проверены терморегулятор и термоголовка. This can sound more like As for being already checked — the thermostat and thermostatic head are.

The original order is the most straightforward.

Yes, that is also correct.

The difference is mostly one of nuance:

• уже проверены
  Often sounds like a current result/state: they are already in the checked state.

• уже были проверены
  More explicitly points to a past action/event: they were already checked.

In many real contexts, both work. The version without были is very common and natural, especially in short technical or status-style statements.

Yes. Russian word order is flexible, although different orders can sound more or less neutral.

The version you have:

• Терморегулятор и термоголовка уже проверены.

is neutral and clear: first the topic, then the comment about them.

Other possible orders include:

• Уже проверены терморегулятор и термоголовка.
• Проверены уже терморегулятор и термоголовка.

These are possible, but they shift emphasis. A learner should usually stick with the original order unless there is a specific reason to highlight already or the fact of being checked.

Then the short participle would agree with that single noun in gender and number:

• Терморегулятор уже проверен. — masculine singular
• Термоголовка уже проверена. — feminine singular

That makes this sentence a useful model:

• masculine singular → проверен
• feminine singular → проверена
• neuter singular → проверено
• plural → проверены

So the plural проверены in your sentence is there because the subject has two items.