1)
Yeah, that's right. My bad. It should be affirmative. その火事を消火するのに時間がかかりました is more natural, though. (I presumed that you must be able to understand the sentence easily if 時間がかかりました follows その火事を消火するのに. So I thought this clause might be put preceding it.)
When ~のに refers to the purpose of a verb/an adjective, only the present tense is used for the verb preceding ~のに. Verbs/Adjectives that can be used for this structure are not so many (e.g. かかる, 要る, 必要だ, 大切だ, 便利だ, 不便だ, 役に立つ, いい, 使う, 利用する, 苦労する,,,). These verbs/adjectives take に as the "purpose" marker also for nouns (e.g. アメリカ旅行にお金が要る, 勉強に使う, 子育てに苦労する,,,), so it's considered that that is from "a nominalizer の + に which is originally needed for the verb/adjective". This noun clause doesn't express "did something" or "not to do something" but just expresses the concept of the verb just like the noun form of it. That's the reason only the present form is used in this structure.
e.g.
消火するのに時間がかかりました = 消火に時間がかかりました
アメリカに旅行するのにお金が要った = アメリカ旅行にお金が要った
勉強するのに使った = 勉強に使った
子供を育てるのに苦労した = 子育てに苦労した
Unlike this ~のに for the porpose, ~のに for adversative conjunction can take the past form of a verb/an adjective (e.g. 風邪を引いていたのに海に行った).
2)
~にちがいない means "must be/should be", thus, this is almost the same as the copula ~だ (with must/should). On the other hand, ちがわない means "to be not different", and に is hardly used to indicate the target to be compared. と is usually used for it (e.g. ひらめきは思いつきと違わない = ひらめきと思いつきは同じだ). Thus, the answer is no, both in the meaning and for the use of particle.