Learn Hardcore Icelandic: Ég elda grænmetið heima.

In Icelandic, the definite article is a suffix, not a separate word. For neuter singular nouns the suffix is -ið. So you have:
• grænmeti (indefinite “vegetables” as a mass noun)
• grænmetið (definite “the vegetables”)

Why is grænmeti singular if it means “vegetables” (plural in English)?

grænmeti is a collective or mass noun in Icelandic—it refers to “vegetables” as a category. It behaves grammatically as a singular noun, so verbs and adjectives agree in the singular.

What case is grænmetið in, and why does it look the same as the nominative?

Here grænmetið is in the accusative case (direct object of elda). Neuter singular nouns have identical forms in the nominative and accusative, so you don’t see a change. The only visible marking is the definite suffix -ið.

Why is there no preposition before heima, and what exactly is heima?

heima is an adverb meaning at home. Icelandic doesn’t require a preposition here—heima on its own expresses location. It derives from the noun heimur (“home/world”) but functions like any other adverb of place.

Can appear somewhere besides the end of the sentence?