Learn Finnish: Jokainen elämänvaihe on yhä tärkeämpi. - Every life stage is increasingly important.

By itself, tärkeämpi just means "more important". When you add yhä in front of a comparative, it indicates continuous change. Together, yhä tärkeämpi translates to "increasingly important" or "more and more important".

Could I use a word like enemmän (more) instead of yhä to mean "increasingly"?

No, to say "more and more" or "increasingly" with an adjective in Finnish, you don't use enemmän. You must use yhä (or aina vain) plus the comparative form of the adjective itself (tärkeämpi).

Why does the compound word elämänvaihe use the genitive form elämän for its first half?

In Finnish compound nouns, the first part is often in the genitive case to show what the second part belongs to or is related to. Elämänvaihe literally means "phase of life", so elämä (life) takes the genitive -n ending.

Since jokainen refers to all life stages, why are elämänvaihe and the verb on in the singular?

The pronoun jokainen ("every" or "each") always takes a singular noun and a singular verb. This works exactly like English, where we say "every stage is" rather than "every stages are".