Veřejný úřad je otevřený.

• veřejný = public
• úřad = office, authority, or government office
• je = is
• otevřený = open

So the structure is:

• veřejný úřad = public office
• je otevřený = is open

Because they both agree with úřad, which is:

• masculine
• inanimate
• singular
• nominative

In Czech, adjectives change their endings to match the noun they describe.

Since úřad is masculine singular, the adjective form is -ý:

• veřejný úřad
• úřad je otevřený

If the noun were feminine, neuter, or plural, the adjective ending would change.

In standard Czech, you normally keep je in a sentence like this.

• Veřejný úřad je otevřený. = correct standard sentence

Unlike Russian, Czech usually does not omit the present-tense form of to be in ordinary statements.

You may sometimes hear shortened or headline-style language without it, but for normal learning and normal speech, use je.

It is nominative singular.

That is because úřad is the subject of the sentence:

• The public office is open.

In Czech, the subject of a basic sentence is usually in the nominative.

For a learner, the simplest way to treat it here is as an adjective meaning open.

In practice, forms like otevřený come from a participial pattern, but in a sentence like this it behaves like an ordinary adjective:

• otevřený úřad = an open office
• úřad je otevřený = the office is open

So the important thing is that it agrees with the noun like an adjective.

Because Czech has no articles.

English says:

• the public office
• a public office

Czech simply says:

• veřejný úřad

Whether it means a or the depends on context.

Yes, it is understandable and natural enough, but in real life Czech often uses just úřad when the context already makes it clear that it is a public/government office.

Depending on context, you might also see:

• státní úřad = state office / government office
• městský úřad = municipal office
• obecní úřad = local council office
• úřad by itself = office / authority

So veřejný úřad works, but Czech often prefers more specific expressions.

Here it means open in the sense of open and operating / available to the public.

Czech uses otevřený for both:

• physically open
• open for visitors, customers, or the public

Context tells you which meaning is intended.

For offices, shops, museums, etc., je otevřený usually means it is open in the practical everyday sense.

That would not mean exactly the same thing.

• Úřad je otevřený. = The office is open.
• Je otevřeno. = It is open. / Open.

A very common Czech way to talk about opening status is with otevřeno:

• Na úřadě je otevřeno. = The office is open.
  Literally something like At the office, it is open.

So yes, Czech often uses je otevřeno in real life, especially on signs or when talking about opening hours. But your original sentence with otevřený is also grammatically correct.

Yes. Czech word order is flexible, because endings carry a lot of grammatical information.

The neutral order is:

• Veřejný úřad je otevřený.

But you could also say:

• Úřad je otevřený.
• Otevřený je veřejný úřad.
• Veřejný je úřad. — but this changes the focus and sounds more marked

The basic meaning may stay similar, but the emphasis changes. For learners, the original word order is the safest and most neutral.

A rough guide for English speakers:

• veřejný ≈ VEH-rzheh-nee
• úřad ≈ OO-rzhat
• je ≈ yeh
• otevřený ≈ OH-tev-rzheh-nee

A few important pronunciation notes:

• ř is a special Czech sound. It is difficult for English speakers. It sounds a bit like a mix of r and zh.
• ú is a long oo
• ý is a long ee
• Czech stress is usually on the first syllable of the word:
  • VE-řejný
  • Ú-řad
  • O-tevřený

Because the sentence needs the subject form, which is the nominative:

• úřad = nominative singular

Other forms appear in other grammatical roles, for example:

• do úřadu = to the office
• z úřadu = from the office
• na úřadě = at the office

In your sentence, the office is the thing being described, so nominative úřad is correct.

Yes, but that is a bit different in style.

• otevřený sounds like a normal adjective: open
• otevřen is a shorter, more participial form, and can sound a little more formal, literary, or passive-like

For most learners, je otevřený is the easier and safer form to use in everyday Czech.

No. In Czech, grammatical gender is not the same as real-life gender.

Úřad is grammatically masculine, but it refers to a thing/institution, not a male person.

That is why the adjectives take masculine forms, even though there is no person involved.

The adjective endings would change to match the noun.

For example:

• Veřejná budova je otevřená.
  (budova = building, feminine)

• Veřejné muzeum je otevřené.
  (muzeum = museum, neuter)

So in your sentence:

• veřejný and otevřený are masculine singular forms because of úřad

That kind of agreement is very important in Czech.