Adresa trvalého pobytu je v registru obyvatel.

Because adresa is followed by a phrase meaning of permanent residence, and in Czech that of ... relationship is usually expressed with the genitive case.

• basic form: trvalý pobyt = permanent residence
• after adresa: adresa trvalého pobytu = address of permanent residence

Both words change into the genitive singular:

• trvalý → trvalého
• pobyt → pobytu

So the phrase literally works like address of permanent residence.

Adresa is in the nominative singular.

It is the subject of the sentence:

• Adresa trvalého pobytu = the address of permanent residence
• je = is
• v registru obyvatel = in the register of inhabitants

So the structure is basically:

• [subject] + [is] + [location]

That is why adresa stays in its dictionary form.

Here je is the 3rd person singular of být (to be), meaning is.

In Czech, the verb to be is normally not omitted in ordinary present-tense statements like this one.

So:

• Adresa ... je v registru obyvatel. = The address ... is in the register of inhabitants.

You cannot normally say:

• Adresa trvalého pobytu v registru obyvatel.

That would sound incomplete in standard Czech.

Because the preposition v usually takes the locative case when it means in or inside a place.

Here the meaning is location, not motion:

• v registru = in the register

The noun registr changes to the locative singular:

• nominative: registr
• locative: registru

So v registru is exactly what you would expect after v for location.

In registru obyvatel, the word obyvatel means of inhabitants / residents.

It is functioning as a dependent noun after registr, so it is in the genitive plural:

• registr obyvatel = register of inhabitants

The reason it may look unchanged is that for this noun, the genitive plural form is the same as the basic form:

• nominative singular: obyvatel = inhabitant
• genitive plural: obyvatel = of inhabitants

So the form stays the same, but the grammar is different.

Yes, it is a common official or administrative expression.

• registr = register, official database
• obyvatel = inhabitants, residents

So registr obyvatel means something like:

• the population register
• the register of inhabitants
• the residents’ register

In legal or bureaucratic Czech, this kind of noun + genitive phrase is very common.

Czech has no articles like a or the.

So:

• Adresa can mean an address or the address
• registru can mean a register or the register

The exact meaning depends on context.

In this sentence, English would normally use the:

• The address of permanent residence is in the register of inhabitants.

But Czech does not need a separate word for that.

Yes. Czech word order is fairly flexible, and changing it usually changes the focus or emphasis, not the core meaning.

For example:

• Adresa trvalého pobytu je v registru obyvatel.
  Neutral statement; the topic is the address of permanent residence.

• V registru obyvatel je adresa trvalého pobytu.
  This puts more emphasis on where it is.

The sentence you have is the most neutral and straightforward version.

It is very often an official legal/administrative term in Czech.

• trvalý pobyt = permanent residence
• adresa trvalého pobytu = address of permanent residence

In everyday life, people may also talk about where someone actually lives, which is not always the same as their official trvalý pobyt.

So this phrase often has a bureaucratic meaning, not just a general one.

Because trvalého does not describe adresa. It describes pobytu.

The structure is:

• adresa = address
• trvalého pobytu = of permanent residence

So the adjective trvalého agrees with pobytu, not with adresa.

Since pobyt is masculine singular and in the genitive, the adjective must also be masculine singular genitive:

• trvalý pobyt
• trvalého pobytu

If trvalý were describing adresa, it would have to be feminine, such as trvalá.

Yes:

• Adresa = address
• trvalého = of permanent
• pobytu = residence / stay
• je = is
• v = in
• registru = register
• obyvatel = inhabitants / residents

Natural English:

• The address of permanent residence is in the register of inhabitants.

Or, in a more official-sounding translation:

• The permanent residence address is recorded in the population register.

Yes. Although the Czech literally looks like address of permanent residence, a more natural English rendering is often:

• permanent residence address
• registered address
• official address of permanent residence

So the Czech phrase is grammatically built as noun + genitive phrase, while English often prefers a compound expression before the noun.