Learn Russian: Происхождение чьих антикварных вещей они обсуждают? - The origin of whose antique items are they discussing?

It matches the case and number of the noun it modifies, вещей (items). Since we are saying the origin 'of' the antique items, антикварных вещей is in the genitive plural. Therefore, the possessive pronoun чей (whose) must also take the genitive plural form, чьих.

How do we get вещей from the dictionary form вещь?

Вещь is a feminine noun ending in a soft sign (-ь), which makes it a 3rd declension noun. For these nouns, the genitive plural ending is always -ей.

Why isn't the question word чьих at the very beginning of the sentence?

In Russian, question words generally come first, but here чьих антикварных вещей acts as a modifier for происхождение (origin). Because they are discussing the origin (the main focus and direct object), происхождение takes the lead to keep the related phrase intact, similar to the English structure.

What case is происхождение in, and why?

It is in the accusative case because it is the direct object of the verb обсуждают (they are discussing). Since происхождение is a neuter inanimate noun, its accusative form looks exactly the same as its nominative dictionary form.