Learn Danish: Hun tror at det gamle hus ikke findes. - She believes that the old house does not exist.

You are right that -s is usually the passive ending in Danish. However, findes is a deponent verb. This means it has a passive form (ending in -s) but an active meaning. While at finde means "to find", the s-form at findes means "to exist".

Why does ikke come before the verb findes here?

This happens because at det gamle hus ikke findes is a subordinate clause (a dependent clause introduced by at). In Danish main clauses, ikke comes after the first verb. But in subordinate clauses, adverbs like ikke must come before the verb.

Why is it gamle instead of gammel?

Because the phrase is definite ("the old house"). When you have a definite article (det) and a noun (hus), any adjective in between must take its definite form, which usually means adding an -e. So, gammel becomes gamle. Notice that the double m drops, which is a specific spelling rule for this word.

Could I use synes or tænker instead of tror here?

In this case, tror is the right choice because she is expressing a belief about a fact (whether the house exists or not). You use synes for subjective opinions (like "I think the house is pretty") and tænker for the active process of using your brain to think.