Learn Finnish: Pelosta ja tästä esteestä huolimatta me jatkamme eteenpäin. - Despite the fear and this obstacle, we continue forward.

Yes. The postposition huolimatta (despite) requires the elative case. When you link multiple nouns with ja (and) before a postposition, every single noun in that list—and any adjectives or pronouns attached to them, like tämä becoming tästä—must be put into that required case.

The word huolimatta looks like a verb form ending in -matta. Is it?

Exactly! It is the abessive form (meaning "without doing") of the verb huolia (to care, to mind). Literally, the phrase means "without caring about the fear..." Over time, it froze into a standard postposition meaning "despite" or "regardless of". This origin is also why it forces the preceding words into the elative case, just as the verb huolia does.

We use jatkamme here. Why wouldn't we use a form of the verb jatkua?

Finnish strictly separates transitive verbs (someone does something) from intransitive verbs (something happens on its own). Jatkaa is active: we (me) are the ones deliberately continuing our movement. Jatkua is only used when a thing continues by itself, for example: Sade jatkuu (The rain continues).

What is the difference between eteen and eteenpäin?