At the very bottom of the following website the author gives an example of a correct sentence:
Das Haus, worin ich wohne, ist neu.
But why is it not:
Das Haus, worin wohne ich, ist neu.
I thought the verb is always second.
And in addition, could someone please explain the effect words such as damit, darauf, darin, womit, and hinaus have on the word order please?
In German you have "Hauptsatz" (main clause) and there the conjugated verb is always in second place and then you have "Nebensatz" (subordinate clause) there the verb is always in the last place.
So for your sentence the "Hauptsatz" is "Das Haus ist neu" and then you have an inserted "Nebensatz" that describes the house further "worin ich wohne"
Also the "wo-words" work in a similar manner. They have the same function for question words and relative pronouns like the da-words for demonstrative pronouns or as conjunctions.