Learn Russian: Все стремятся жить лучше.

Все literally means "all" and is used collectively, which is why the verb стремятся is in the plural form. Каждый means "every" or "each person" individually, and would require a singular verb. Both are correct, but Все is very natural for broad generalizations.

The current lesson is about verb government. How does the verb стремиться govern other words?

In this sentence, it simply links to an infinitive verb (жить). However, when you strive for a noun (like a goal or success), стремиться requires the preposition к followed by the Dative case. For example, стремиться к успеху means "to strive for success."

Why does стремятся have the reflexive -ся ending? Are they striving "themselves"?

No. While the -ся ending often means an action is done to oneself, стремиться is a verb that simply doesn't exist without it. At the B1 level, you will notice several of these "inherently reflexive" verbs that have a standard, non-reflexive meaning in English.

Is лучше an adjective or an adverb in this sentence?

Here, it acts as an adverb because it describes the verb жить (how do they want to live? better). In Russian, the comparative form лучше is exactly the same whether it is used as an adjective or an adverb.