Learn Hardcore Russian: Эта ситуация совершенно новая для меня.

Because ситуация is a feminine noun in the nominative case.
– эта is the feminine singular demonstrative pronoun (“this”).
– это is the neuter singular demonstrative pronoun and wouldn’t agree with a feminine noun.

Why is новая used here, and not новый or новое?

Adjectives in Russian must agree with the noun they modify in gender, number, and case.
– новая is the nominative singular feminine form of новый.
– новый is masculine, новое is neuter, and новые is plural.

What is совершенно, and why is it placed before новая?

совершенно is an adverb meaning “completely” or “absolutely.” It modifies the adjective новая, intensifying it.
– Word order in Russian is relatively flexible, but adverbs of degree typically precede the adjective they modify: совершенно новая.

What case is для меня, and why?

для is a preposition that always requires the genitive case.
– меня is the genitive form of я (“I”).
So для меня literally means “for me.”

Could the sentence be reordered, for example “Для меня эта ситуация совершенно новая”?

Yes. Russian word order is quite flexible. Starting with Для меня shifts the emphasis onto the fact that it’s new for me.
– Neutral word order (subject–predicate): Эта ситуация совершенно новая для меня.
– Emphatic: Для меня эта ситуация совершенно новая.

Are there synonyms or alternative intensifiers for совершенно?

Yes, you can use other adverbs to intensify новая:
– абсолютно новая (absolutely new)
– полностью новая (fully new)
– крайне новая (extremely new, less common)
Or you might omit the intensifier and say simply Эта ситуация новая для меня.