Learn Hardcore German: Zur Hochzeit schenkt mir meine Mutter eine Kette aus Leder. - For the wedding my mother gives me a necklace made of leather.

They mean the same thing; zur is just the standard, smoother form in everyday German.
You could say zu der Hochzeit, but it would sound more formal/emphatic or a bit stiff in this sentence.

Why is “Hochzeit” in the dative case, and how can I tell?

The preposition zu always takes the dative case.

Nominative: die Hochzeit
Dative: der Hochzeit

In the sentence, (zu)r Hochzeit = zu der Hochzeit → dative.
So any noun after zu will be dative:

zu dem Haus (zum Haus)
zu der Schule (zur Schule)
zur Hochzeit

Why is there no article before “Hochzeit” that I can see?

There is an article, but it’s hidden inside the contraction:

zu der Hochzeit → zur Hochzeit

So the article der (dative feminine) is still there, just fused with zu.

Could the sentence also start with “Meine Mutter”? What changes then?

Yes:

Zur Hochzeit schenkt mir meine Mutter eine Kette aus Leder.
Meine Mutter schenkt mir zur Hochzeit eine Kette aus Leder.

Both are correct and mean essentially the same.
What’s happening:

In German main clauses, the finite verb must be in 2nd position.
You can put a different element in first position (time, place, subject, etc.), and the verb stays second.

So:

1st version: Zur Hochzeit (1st), schenkt (2nd), then the rest.
2nd version: Meine Mutter (1st), schenkt (2nd), then the rest.

The first version just emphasizes the occasion a bit more; the second emphasizes the subject (“my mother”) more naturally.

Why is it “schenkt mir meine Mutter” and not “schenkt meine Mutter mir”?

Both word orders are grammatically correct:

… schenkt mir meine Mutter eine Kette aus Leder.
… schenkt meine Mutter mir eine Kette aus Leder.

General tendencies:

Pronouns (like mir) usually come before full noun phrases (like meine Mutter).
So mir meine Mutter follows the usual pattern (pronoun) + (noun).

However, in actual speech, many people would more naturally say:

Meine Mutter schenkt mir zur Hochzeit eine Kette aus Leder.

Your original sentence uses a slightly marked word order, but it’s still perfectly correct.

Why is it “mir” and not “mich”?

Because “mir” is dative, and the verb schenken uses:

a dative object = the person receiving the gift
an accusative object = the thing that is given

Forms of ich:

Nominative: ich (subject)
Accusative: mich (direct object)
Dative: mir (indirect object)

In this sentence:

meine Mutter = subject (who gives)
mir = dative (to whom she gives)
eine Kette aus Leder = accusative (what she gives)

So mir is correct: “She gives me (indirect object) a necklace.”

What is the difference between “schenken” and “geben”?

Both involve giving, but:

geben = to give (very general)
- Meine Mutter gibt mir eine Kette. – My mother gives me a necklace.
schenken = to give as a gift (usually for an occasion)
- Meine Mutter schenkt mir eine Kette. – My mother is giving me a necklace as a present.

Because of “zur Hochzeit” (for the wedding), schenken is the natural choice: it’s clearly a gift.

Why is it “eine Kette” and not “ein Kette”?

Because Kette is feminine in German:

die Kette (singular)
eine Kette (indefinite article, nominative/accusative)

Articles by gender:

Masculine: ein Mann
Neuter: ein Kind
Feminine: eine Frau, eine Kette

So “necklace” → die Kette, and here as a direct object (accusative feminine): eine Kette.

Why “aus Leder” and not “von Leder” or something else?

The preposition aus is used to express the material something is made of:

eine Kette aus Leder – a necklace made of leather
ein Tisch aus Holz – a table made of wood
ein Ring aus Gold – a ring made of gold

In this sense, von is not used. Aus is the standard preposition to show material.

What case is “Leder” in, and why doesn’t it have an article?

“Leder” is:

neuter, and
in the dative case, because it comes after aus, which always takes dative.

Dative forms for “leather”:

Nominative: das Leder
Dative: dem Leder

In practice, with materials and uncountable substances, German often omits the article:

aus Leder (not aus dem Leder)
aus Holz, aus Gold, aus Glas

So “Leder” is dative neuter, but without an article because it’s used as a material.

Does “zur Hochzeit” mean “to the wedding” or “for the wedding”?

In this sentence it means “for the wedding” in the sense of:

On the occasion of the wedding / as a wedding present

So the natural translation is:

“For my wedding, my mother is giving me a leather necklace.”

Zu + dative can mean:

destination: Ich gehe zur Hochzeit. – I’m going to the wedding.
occasion/time: Zur Hochzeit schenkt sie mir … – For the wedding, she is giving me …

Here, the context with schenken makes it clear it’s about the occasion, not movement.

Why are Hochzeit, Mutter, Kette, and Leder capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

Hochzeit (wedding) – noun → capitalized
Mutter (mother) – noun → capitalized
Kette (necklace/chain) – noun → capitalized
Leder (leather) – noun → capitalized

This is a fixed rule in modern German orthography: every noun gets a capital letter.

Is “Kette” always “necklace,” or can it mean something else?

Kette is a general word meaning “chain” or “necklace”, depending on context:

eine Kette am Fahrrad – a chain on the bicycle
eine Halskette – specifically “a necklace” (Hals = neck)
eine Kette aus Gold – often understood as a jewelry chain / necklace, depending on context.

In “eine Kette aus Leder” with a gift context, it will normally be understood as a necklace made of leather. If you want to be explicit, you can say eine Halskette aus Leder.