W urzędzie wypełniam formularz i wpisuję adres.

Not quite.

Urząd usually means an official, public, or government office, such as:

• a city office
• a tax office
• a registry office

Biuro is more like:

• an office as a workplace
• an office room
• a business office

So w urzędzie suggests you are in an administrative/public office, not just in someone's office at work.

Because after w meaning in / at a place, Polish normally uses the locative case.

So:

• basic form: urząd
• after w for location: w urzędzie

This is a very common pattern:

• w domu = in the house
• w sklepie = in the shop
• w urzędzie = in/at the office

The form urzędzie may look a bit irregular at first, but it is just the correct locative singular form of urząd.

Because Polish usually does not need subject pronouns when the verb ending already shows who is doing the action.

Here:

• wypełniam = I fill out / I am filling out
• wpisuję = I enter / I am entering

The -am and -uję endings already tell you the subject is I.

You can say Ja w urzędzie wypełniam formularz..., but ja is usually added only for emphasis, contrast, or clarity.

They are both in the present tense.

More specifically:

• wypełniam = present tense of wypełniać
• wpisuję = present tense of wpisywać

In Polish, the present tense of an imperfective verb can mean:

• something happening right now: I am filling out...
• something done habitually / generally: I fill out...

So this sentence could describe:

• what you are doing at the moment, or
• a routine action

The exact meaning depends on context.

Because Polish verbs belong to different conjugation patterns.

Here:

• wypełniać → wypełniam
• wpisywać → wpisuję

So even though both mean I ..., the ending is not always the same. In Polish, first-person singular present can end in different ways, such as:

• -am
• -ę
• -uję

This is something you gradually learn verb by verb, though patterns do exist.

They are both direct objects, so they are in the accusative case.

However, both nouns are masculine inanimate singular, and for that group the accusative is often identical to the nominative.

So:

• formularz = nominative
• formularz = accusative

and:

• adres = nominative
• adres = accusative

That is why they look unchanged.

Compare this with a masculine animate noun, where the accusative usually does change:

• kot = cat
• mam kota = I have a cat

They describe two slightly different actions.

• wypełniam formularz = I fill out a form
  This means completing the form as a whole.

• wpisuję adres = I enter/write in an address
  This means putting one specific piece of information into the form.

So the sentence has a natural sequence:

1. you fill out the form
2. you enter the address on it

You do not know from the noun alone, because Polish has no articles like a, an, or the.

So:

• formularz can mean a form or the form
• adres can mean an address or the address

Context tells you which one is meant.

That also means w urzędzie could be translated as:

• in the office
• at the office
• sometimes even in an office

depending on the situation.

Yes. Polish word order is fairly flexible.

The given sentence: W urzędzie wypełniam formularz i wpisuję adres.

is natural and neutral, with w urzędzie placed first to set the scene: At the office...

Other orders are possible, for example:

• Wypełniam formularz i wpisuję adres w urzędzie.
• Formularz wypełniam i wpisuję adres w urzędzie.

But different word orders can sound more marked, literary, or emphatic. The original version is a good standard one for learners.

This is the difference between location and movement.

• w urzędzie = in / at the office
  You are already there.

• do urzędu = to the office
  You are going there.

Examples:

• Jestem w urzędzie. = I am at the office.
• Idę do urzędu. = I am going to the office.

So in your sentence, w urzędzie is used because the action happens there.

A rough guide:

• urzędzie ≈ oo-ZHEN-jye
• wypełniam ≈ vih-PEW-nyam

A few useful pronunciation notes:

• rz sounds like the s in measure
• ł sounds like English w
• ni before a vowel often sounds like ny
• ę before another consonant is often pronounced more like en/em in everyday speech

So:

• w urzędzie sounds roughly like voo-ZHEN-jyeh when spoken together
• wypełniam has that ł = w sound, so not wypełniam with an English l

The exact pronunciation takes practice, but these rough English guides are a good start.