Learn Russian: Старайтесь упростить эти формальности как можно быстрее. - Try to simplify these formalities as quickly as possible.

It literally translates to "how it is possible faster." Adding как можно before a comparative adverb (like быстрее, meaning "faster") is the standard Russian formula for saying "as [adverb] as possible."

Why do we pair the imperfective verb старайтесь with the perfective verb упростить?

Старайтесь (try/endeavor) is imperfective because it describes an ongoing or general effort. However, the goal of that effort is to completely simplify the paperwork, which is a single, finished result. Therefore, the perfective упростить is used for the action being attempted.

Is формальности in the accusative case here? Why doesn't the ending change?

Yes, it is the direct object, so it is in the accusative plural. Формальность is a feminine inanimate noun ending in a soft sign (-ь). For all inanimate nouns, the accusative plural looks exactly the same as the nominative plural.

Could I use как можно скорее instead of как можно быстрее in this sentence?

Yes, they are almost perfectly interchangeable here. Быстрее (from быстро) focuses slightly more on the physical speed of the process, while скорее (from скоро) focuses on getting it done sooner rather than later. Both translate well to "as quickly/soon as possible."