Её амбиции были огромными, ей пришлось рискнуть ради власти.

Questions & Answers about Её амбиции были огромными, ей пришлось рискнуть ради власти.

The lesson is about ради. Could I just use для here instead, and what case does ради trigger?
You could use для власти (for power), but ради translates closer to 'for the sake of' and adds an emotional or high-stakes emphasis to the ultimate purpose. Both ради and для require the genitive case, which is why the feminine noun власть changes to власти.
Why is the adjective огромными in the instrumental case rather than the nominative огромные?
When using the past tense verb быть (here, были), predicate adjectives are very often put in the instrumental case to describe a state or characteristic. You will sometimes see the nominative, but the instrumental is usually preferred in standard and written Russian.
Why use the impersonal ей пришлось instead of она должна была for 'she had to'?
ей пришлось implies that external circumstances forced her to take the risk; she felt she had no other choice to get what she wanted. она должна была would imply a moral obligation, duty, or a pre-arranged plan, which does not fit the context of taking a risk for power.
Why does the noun власть take the ending -и in the genitive case here?
власть is a feminine noun ending in a soft sign (-ь), meaning it belongs to the 3rd declension. For 3rd declension nouns, the genitive, dative, and prepositional cases all take the -и ending.

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