Learn Hardcore German: Der Hund läuft schnell zum Strand.

läuft is the 3rd person singular present tense form of the verb laufen (to run, and in some regions also to walk). German verbs in the present tense add -t for er/sie/es (he/she/it):

ich laufe
du läufst
er/sie/es läuft

Could I use rennt or geht instead of läuft, and would the meaning change?

Yes, you can replace läuft with other verbs of motion, but the nuance shifts:

rennen (er rennt) means “to sprint” or “dash” – more forceful than laufen.
gehen (er geht) means “to walk” – slower than laufen (in the sense “run”).
laufen is neutral and can mean “run” (standard) or “walk” (colloquially, especially in southern Germany).

Why is schnell not declined like an adjective (e.g. schnelle)?

Here schnell is an adverb modifying the verb läuft (“runs quickly”). Adverbs in German never take adjective endings. Only when they directly modify a noun (as adjectives) would you see endings, for example ein schneller Hund.

What is zum, and how is it formed?

zum is a contraction of zu + dem.

zu is the preposition meaning “to” or “toward.”
dem is the masculine/neuter dative definite article.
Together they form zum, which you use before a masculine or neuter noun in dative case.

Why is Strand in the dative case here?

Because the preposition zu always governs the dative. So whenever you use zu (or its contraction zum), the following noun must be in the dative case:
zu + dem Strand → zum Strand.

Why is Der Hund in the nominative case, and not something like Den Hund?

In German, the subject of a sentence takes the nominative case. Here Der Hund is the subject performing the action (running), so it stays in the nominative. Den Hund would be accusative, used for direct objects, which this is not.

Why does läuft appear in the second position of the sentence?

German main clauses follow the verb-second (V2) rule: the finite verb must occupy position 2. Anything you put first (here Der Hund) is position 1; the verb läuft then comes directly after in position 2.

Why is zu used here instead of another preposition like an, as in an den Strand?

Both zum Strand and an den Strand can express going to the beach, but with a slight nuance:

zu(n) emphasizes general direction toward a place.
an
- accusative (an den Strand) emphasizes arriving at the very edge or border of something (the shore).
  In many contexts they’re interchangeable, but zum Strand is common when focusing on the destination in a broader sense.

Which question word would I use in German to ask “Where is the dog running quickly to?”

For asking “to where?” in a directional context, use wohin. You’d form:
Wohin läuft der Hund schnell?
Here wohin is in position 1, läuft stays in position 2, and der Hund follows.