Je ne connais pas le code postal de cette adresse.

Because French usually uses connaître with a noun phrase, while savoir is more often used with a clause or an infinitive.

• connaître + noun: Je connais ce mot.
• savoir + clause/question/infinitive: Je sais où il habite. / Je sais parler français.

Here, le code postal is a noun phrase, so Je ne connais pas le code postal... is natural.

If you wanted to use savoir, you would normally reshape the sentence:

• Je ne sais pas quel est le code postal de cette adresse.

Connais is the first-person singular present tense form of connaître.

So:

• je connais = I know
• tu connais = you know
• il/elle connaît = he/she knows

A useful thing to notice: the final -s in connais is not pronounced.

Ne ... pas is the standard way to make a sentence negative in French.

The verb here is connais, and the negative wraps around it:

• Je connais le code postal. = I know the postal code.
• Je ne connais pas le code postal. = I do not know the postal code.

So:

• ne goes before the verb
• pas goes after the verb

Not always. In careful writing and formal speech, you should use ne ... pas. But in everyday spoken French, people very often drop ne:

• Je ne connais pas le code postal. = standard
• Je connais pas le code postal. = very common in speech

As a learner, it is best to learn and use the full form first, even if you later hear the shorter spoken version.

Because the sentence is talking about a specific postal code: the one that belongs to that address.

• le code postal = the postal code
• un code postal = a postal code

Using le makes sense because an address normally has one particular postal code associated with it.

Here, de links le code postal to cette adresse.

Literally, it is structured like:

• the postal code of this address

In natural English, you might say:

• the postal code for this address

But French commonly uses de in this kind of noun relationship:

• le nom de la rue
• la couleur de la voiture
• le code postal de cette adresse

So even if English often prefers for, French naturally uses de here.

Because adresse is a feminine singular noun, and the demonstrative adjective has to agree with it.

French uses:

• ce for masculine singular before most consonants
• cet for masculine singular before a vowel or silent h
• cette for feminine singular
• ces for plural

Since adresse is feminine singular, you need cette:

• cette adresse

Because cette already functions as the determiner. In French, you do not put a regular article like le / la / les in front of a demonstrative adjective.

So you say:

• cette adresse

Not:

• la cette adresse

And after de, it stays:

• de cette adresse

This is the same pattern as:

• de ce livre
• de cet hôtel
• de cette ville
• de ces documents

Yes. In French, ce / cet / cette / ces can mean either this or that, depending on context.

So cette adresse can mean:

• this address
• that address

If you want to make the distinction clearer, French can add:

• cette adresse-ci = this address
• cette adresse-là = that address

But in many everyday sentences, French just uses cette adresse and lets the context decide.

The structure is:

• Je = subject
• ne ... pas = negation
• connais = verb
• le code postal = direct object
• de cette adresse = complement giving more information about the object

So the sentence is built like this:

Subject + negative marker + verb + negative marker + object + complement

That is why French says:

• Je ne connais pas le code postal de cette adresse.

Even though English may naturally prefer:

• I don’t know the postal code for this address.

A careful pronunciation is approximately:

/ʒə nə kɔnɛ pa lə kɔd pɔstal də sɛt adʁɛs/

A rough English-friendly guide would be:

zhuh nuh ko-NAY pah luh kode pos-TAL duh set ah-DRESS

A few useful pronunciation notes:

• je sounds like zhuh
• connais ends with an ay sound
• the final -s in connais is silent
• the final -e in code is silent
• the final -t in postal is silent
• the final -e in adresse is silent

In fast speech, some of the very small vowel sounds may be reduced.

Yes, functionally it is the same idea: the number or code used for mail delivery.

French normally says code postal, not ZIP code.

A couple of notes:

• In France, a code postal is usually a five-digit number.
• In Canada, French also uses code postal, even though Canadian postal codes include both letters and numbers.

So if you are translating from English, ZIP code is usually best matched by code postal.

Yes. That is also correct and natural.

The difference is mainly in structure:

• Je ne connais pas le code postal de cette adresse.
• Je ne sais pas quel est le code postal de cette adresse.

The second version is a little more explicit because it literally means something like I do not know what the postal code of this address is.

Both are good, but the original sentence is shorter and very natural.