Learn Russian: Ювелир внимательно изучает каждую грань. - The jeweler carefully studies every facet.

Изучать means to study something in depth, investigate, or closely scrutinize it. You use it when talking about researching a subject or examining a physical object, like the jeweler is doing here. Учить is generally used for memorizing (like vocabulary) or learning a specific skill, while учиться means to be a student at an institution.

Why doesn't the ending of грань change here, given that it's the direct object?

Грань is a feminine noun ending in a soft sign (-ь), putting it in the 3rd declension. For 3rd declension inanimate nouns, the accusative case is identical to the nominative case. So even though it's the direct object, it remains грань, while its modifier takes the standard feminine accusative ending (каждую).

Could we use любую or всякую instead of каждую for 'every facet'?

Not without changing the meaning. Каждый means 'each' or 'every' single one of a specific set (the facets of the gem). Любой means 'any' (whichever one you like), and всякий usually means 'every sort of' or 'any kind of'. Since the jeweler is looking at each specific facet in sequence, каждую is the correct choice.

Besides 'facet', how else is the word грань used in Russian?