Learn Arabic (Modern Standard): ليس هذا النعل زلقا.

The word ليس (is not) is a special verb that negates nominal sentences. Like كان (was), it leaves the subject in the nominative case but changes the predicate (the describing word) to the accusative case. Since زلقا is the predicate here, it takes the accusative ending -an, which is written with an extra Alif at the end.

Since هذا (this) is the subject of ليس, why does the next word النعل (the sole) take the nominative case with a Damma?

When a demonstrative pronoun like هذا is immediately followed by a definite noun (a noun starting with ال), they form a single phrase meaning "this [noun]". Grammatically, النعل copies the case of the pronoun it follows. Because هذا is in the subject position of ليس (which requires the nominative case), النعل also takes the nominative case, shown by the Damma on its last letter.

Could I use the preposition بِ (bi) with the predicate here?

Yes, absolutely! It is very common in Arabic to attach the preposition بِ to the predicate of ليس. If you do that, the word is forced into the genitive case by the preposition, so it becomes بزلقٍ (bi-zaliqin) with a double Kasra, rather than the accusative زلقا (zaliqan). Both structures mean the exact same thing.

Why is the negating word ليس at the very beginning of the sentence?