Learn Hardcore Russian: Финансовый отчёт, написанный вчера, готов. - The financial report written yesterday is ready.

In Russian, the subject is in the nominative case, so отчёт is nominative singular masculine, and финансовый agrees with it in gender, number, and case. Everything else in the sentence either describes this subject (написанный вчера) or says something about its state (готов).

What exactly is написанный? Is it a verb, an adjective, or something else?

Написанный is a past passive participle formed from the perfective verb написать (to write, to have written).

написать → написанный = (having been) written

Grammatically, participles behave like adjectives: they agree in gender, number, and case with the noun they modify. So написанный describes отчёт and must be masculine, singular, nominative: написанный отчёт = the report (that was) written.

Why is the phrase написанный вчера set off by commas: Финансовый отчёт, написанный вчера, готов?

The participial phrase написанный вчера is detached (non-restrictive) information that adds an extra detail about the report. The core message is:

Финансовый отчёт готов. – The financial report is ready.

By adding , написанный вчера, you are giving additional information: the report (which was) written yesterday. In Russian, a participial phrase that comes after the noun it describes is normally separated by commas when it is extra, descriptive information rather than strictly necessary to identify which report is meant. Hence the two commas around написанный вчера.

Could we say Финансовый отчёт который был написан вчера готов instead of using написанный вчера?

You can express the same idea with a relative clause, but it must be punctuated correctly:

Финансовый отчёт, который был написан вчера, готов.

This is grammatically correct and means the same thing.
Differences:

написанный вчера is more compact and stylistically a bit more formal or written.
который был написан вчера is slightly heavier and more explicit, closer to English which was written yesterday.

Both are acceptable; participles are simply a more concise way to say it.

Why is написанный masculine and not something like написанное or написанная?

На-писан-ный must agree with отчёт in gender, number, and case.

отчёт – masculine, singular, nominative
So: написанный отчёт – masculine, singular, nominative participle

Other forms would be used with different nouns:

написанная статья (fem.) – the article that was written
написанное письмо (neut.) – the letter that was written
написанные отчёты (pl.) – the reports that were written

Why do we use готов and not готовый in this sentence?

Готов is the short-form adjective, and it works like a predicate: it states the current state of the subject.

Отчёт готов. – The report is ready.

Готовый is the long-form adjective, and it is mainly used before a noun to describe it:

готовый отчёт – a ready/completed report

In predicative position (after the subject), Russian usually prefers the short form: Финансовый отчёт … готов, not … готовый.

Why is there no explicit verb like есть (is) between отчёт and готов?

In Russian, the present tense of быть (есть) is normally omitted in simple statements like X is Y.

English: The report is ready.
Russian: Отчёт готов. (not Отчёт есть готов.)

The structure is simply [subject] + [short-form adjective] to mean X is Y. So Финансовый отчёт … готов naturally means The financial report is ready.

Can we change the word order, like Финансовый отчёт готов, написанный вчера or Готов финансовый отчёт, написанный вчера?

Possible variants and their feel:

Готов финансовый отчёт, написанный вчера.
– Grammatically fine, more stylistic/emphatic: starts with готов, stressing that the readiness is important.
Финансовый отчёт готов, написанный вчера.
– This sounds awkward or incorrect: написанный вчера is left hanging at the end and does not integrate naturally into the structure. Native speakers would normally avoid this.

The original word order Финансовый отчёт, написанный вчера, готов. is the most neutral and natural for standard written Russian.

Could we say вчера написанный финансовый отчёт готов without commas?

Technically, вчера написанный финансовый отчёт готов is possible, but it sounds heavy and somewhat awkward in normal speech. Also, in standard punctuation:

A participial phrase after the noun is usually set off by commas.
A participial phrase before the noun (as a tight attribute) is more likely to be written without commas, but it still must sound natural.

Here, вчера написанный финансовый отчёт feels overloaded as a pre-noun phrase, so the original version with a detached participial phrase and commas is more natural:
Финансовый отчёт, написанный вчера, готов.

Why do we use the participle from написать (perfective) and not from писать (imperfective), like пишущийся вчера?

The verb написать (perfective) presents the action as completed, which matches the idea that the report is now done and ready.

написанный отчёт – a (fully) written report

An imperfective participle like пишущийся would describe an ongoing process:

пишущийся отчёт – a report that is being written (still in progress)

Since the sentence is about a finished product that is now ready, the perfective написанный is the natural choice.