Learn Icelandic: Tilraunin var ómöguleg fyrir mörgum árum síðan. - The experiment was impossible many years ago.

The preposition fyrir in the fyrir ... síðan structure always forces the time period into the dative case. Mörgum árum is the dative plural form of mörg ár (many years).

Why doesn't the adjective ómöguleg end in -t (ómögulegt)?

Adjectives must agree in gender and number with the noun they describe, even when they come after the verb "to be" (predicate adjectives). Tilraun (experiment) is a feminine noun, so it takes the feminine nominative singular form ómöguleg. If we used a neuter noun like verkefnið (the project), it would be ómögulegt.

Since this lesson covers the past subjunctive, why is var in the indicative, and what would the subjunctive look like?

The indicative var is used here because it is stating a concrete fact: the experiment actually was impossible. If you wanted to express a hypothetical, like "The experiment would be impossible," you would use the past subjunctive form væri. Vera is an irregular verb, so it doesn't follow the weak verb endings you are currently learning, but the logic of swapping from indicative to subjunctive for hypotheticals is exactly the same.