Код доступа и навесной замок уже проверили.

Because the sentence has a compound subject: код доступа и навесной замок — the access code and the padlock.

In Russian, when two singular nouns are joined by и (and), the verb normally goes in the plural:

• Код доступа уже проверили. — The access code has already been checked.
• Навесной замок уже проверили. — The padlock has already been checked.
• Код доступа и навесной замок уже проверили. — The access code and the padlock have already been checked.

So проверили is plural because there are two things being checked.

In the past tense, Russian verbs agree in gender/number, not person.

For проверить (to check, perfective), the past forms are:

• проверил — masculine singular
• проверила — feminine singular
• проверило — neuter singular
• проверили — plural

Since the subject here is plural, the verb is проверили.

Russian often leaves out the doer when it is unknown, unimportant, obvious, or generic.

So проверили here can mean something like:

• they checked
• someone checked
• it has been checked / they have been checked

This is a very common Russian pattern. The sentence does not explicitly say who did the checking.

So even though the verb form is literally they checked, the natural English translation may be more passive-like:

• The access code and the padlock have already been checked.

Код доступа is a standard Russian noun phrase meaning access code.

The second noun, доступа, is in the genitive case and specifies what kind of code it is:

• код — code
• доступ — access
• код доступа — access code

This is a very common Russian structure:

• чашка чая — a cup of tea
• центр города — the city center
• номер телефона — phone number
• код доступа — access code

English often uses a noun-noun structure (access code), but Russian often uses noun + genitive noun.

Навесной замок means padlock.

Literally:

• замок — lock
• навесной — hanging / attached-on / padlock-type

So навесной замок is the standard way to say padlock in Russian.

Be careful with замок, because it can mean either:

• замо́к — lock
• за́мок — castle

The meaning is clear here because of навесной.

Уже means already.

So:

• проверили — checked
• уже проверили — already checked / have already checked

It shows that the action was completed before now or earlier than expected / relevant.

Compare:

• Код доступа и навесной замок проверили. — The access code and the padlock were checked.
• Код доступа и навесной замок уже проверили. — The access code and the padlock have already been checked.

Yes, Russian word order is relatively flexible.

This sentence is perfectly natural, but you could also say:

• Код доступа и навесной замок проверили уже.
• Уже проверили код доступа и навесной замок.
• Код доступа уже проверили, и навесной замок тоже.
  (slightly different structure)

The version you have is neutral and natural.

Very roughly:

• putting уже before the verb is common and neutral
• moving words around can change focus or emphasis, not the basic meaning

So the sentence sounds like a normal statement of fact.

This is about aspect.

• проверить → проверили = perfective
• проверять → проверяли = imperfective

Here проверили is used because the sentence presents the checking as a completed result:

• They have checked / They checked it already.

If you said проверяли, it would suggest more of a process, repetition, or background action:

• they were checking
• they used to check
• they checked (as an activity, not focusing on completion)

Since уже often goes naturally with a completed result, проверили makes very good sense here.

Yes.

They are the things that were checked, and in this sentence they function as the grammatical subject of the clause, so they appear in the nominative:

• код — nominative singular
• замок — nominative singular

Inside код доступа, the word доступа is genitive, but that is because it depends on код:

• код = nominative
• доступа = genitive dependent noun

So the whole phrase код доступа acts as one nominative subject phrase.

Yes. Russian often prefers active-looking sentences with an omitted agent, like the one here, but a passive version is also possible.

For example:

• Код доступа и навесной замок уже проверены.

This means:

• The access code and the padlock have already been checked.

Difference in feel:

• Код доступа и навесной замок уже проверили.
  Literally they already checked...; very natural conversational Russian.
• Код доступа и навесной замок уже проверены.
  More explicitly passive; also natural, sometimes a bit more formal or result-focused.

Both are correct, but the original sentence is very common Russian style.

Because и is simply connecting two items in a list:

• код доступа
• навесной замок

When Russian joins two homogeneous parts with a single и, no comma is used:

• мама и папа
• чай и кофе
• код доступа и навесной замок

A comma would only appear in different kinds of structures, not in a simple A and B combination like this.