Наши обязанности требуют столько же энергии, сколько и ваши.

Questions & Answers about Наши обязанности требуют столько же энергии, сколько и ваши.

Why is энергии in the Genitive case here?
It is controlled by the word столько (so much / as much). Words denoting quantity like столько and сколько require the noun that follows to be in the Genitive case. Since "energy" is uncountable, we use the Genitive singular (энергии).
What is the purpose of же and и in столько же ... сколько и? Can I drop them?
You can drop them and the grammar would still be correct, but they are extremely common in these comparisons. The particles же and и are added to emphasize exact equivalence. Together, they make the sentence mean "just as much energy as yours" rather than a simple "as much energy as yours."
Why does the verb требовать (to require) conjugate to требуют, losing the -ова- part?
This is a standard verb shift that is easy to forget. Verbs ending in -овать or -евать drop the -ова- or -ева- and replace it with -у- or -ю- in all present-tense conjugations. Therefore, требовать becomes я требую, он требует, and они требуют.
Why is the final word ваши (yours) in the Nominative case instead of matching the Genitive of энергии?
Because ваши does not modify энергии. It refers back to the subject of the sentence, обязанности (duties), which is in the Nominative plural. The end of the sentence is essentially a shortened version of "...as your duties (require)."

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