Learn German: Sie bekommen die Katze, sofern Sie die Zustimmung des Vermieters nachweisen. - You get the cat, provided that you prove the consent of the landlord.

While wenn simply means "if", sofern is more formal and restrictive. It translates closer to "provided that" or "as long as". It emphasizes that getting the cat is strictly dependent on proving the landlord's consent.

Why is the separable verb nachweisen written as a single word at the end?

Because sofern is a subordinating conjunction, it kicks the conjugated verb to the very end of the clause. When a separable verb like nachweisen is pushed to the end, its prefix (nach-) and core verb (weisen) rejoin into one word.

Why does Vermieter have an -s at the end to make des Vermieters?

This is the genitive case, which indicates possession or "of the". To say "the consent of the landlord", the masculine noun Vermieter takes the genitive article des and adds an -s to the end of the noun.

Could I use vorausgesetzt, dass instead of sofern here?

Yes! vorausgesetzt, dass (provided that) carries the exact same restrictive condition as sofern. You could easily say: ...vorausgesetzt, dass Sie die Zustimmung des Vermieters nachweisen. Both are excellent B2-level choices.