Learn German: Was für ein Tier lebt diesseits des Hanges? - What kind of animal lives on this side of the slope?

The preposition diesseits (on this side of) requires the genitive case. Since der Hang (the slope) is a masculine noun, its genitive form becomes des Hanges.

Could I write des Hangs instead of des Hanges?

Yes, you can. For many one-syllable masculine and neuter nouns, the genitive ending can be either -s or -es. Both des Hangs and des Hanges are correct, but the -es ending often flows a bit better and is very common in written German.

Why doesn't the word ein in the phrase was für ein have a case ending here?

The word ein in the fixed question structure was für ein (what kind of) declines to match the case and gender of the noun it is attached to. Because the animal is the subject of the sentence, Tier is in the nominative case. Since it is a neuter noun, the correct nominative form is just ein.

Is diesseits commonly used in everyday spoken German?

It is perfectly understood, but it belongs to a slightly more formal or literary register. In a very casual conversation, a native speaker might instead say auf dieser Seite des Hanges to mean the exact same thing.