Hola Mary

Your answers using the articles are also correct, so we'll be adding those to the exercise.

The answer to your question is unfortunately not a straight one because the presence and omission of articles in Spanish is one of the most difficult things to master. Generally, the omission of the definite articles are associated with the intention of making meanings of the nouns more "generic", as opposed to having a specific "thing" in your mind, so, I can say:

En la granja criaban perros y gatos.

At the farm they raised dogs and cats.

or

En la granja criaban los perros y los gatos.

At the farm they raised the dogs and the cats.

In the first one I don't really have an image in my head of those dogs and cats (omission of the articles), it's a more generic talking.

In the second one using the articles I visualize those animals in my head, as if I know them (maybe because I've been there and seen them?) - I am "identifying" them. 

But this is a general approach to articles in Spanish. In those specific phrases in the writing challenge you did, the nuances are a bit different:

"Aparecerás en el perfil de (la) búsqueda..." - here, we were initially using it without the article because we're treating this "perfil de búsqueda" as a set phrase used in the vocabulary of a website that is about finding an au pair - it's more like something used generically to define a part of the website "perfil de búsqueda" (search profile).

" tu disponibilidad y (el) país preferido..." - here, when omitting the article, we are either not repeating a determinant which is somehow obvious "tu disponibilidad y tu país preferido", or simply "simplifying the idea" by omitting it "tu disponibilidad y país preferido" which is often done especially in slightly more formal contexts, like this one, which is explaining some kind of procedure when using the web. 

"Como au pair" - here, we are seeing "au pair" as a profession (even if it is something temporary)  and there is a tendency to omit the articles when stating professions unless there is a lot more detail in the sentence, like descriptive adjectives for example or subordinate clauses related to the profession, e.g. "Como una au pair orgullosa que adora su trabajo...". 

I hope this showed a bit more light on the article topic. We keep searching information from grammarians about this difficult topic but there doesn't seem to be a rule or a specific set of rules that simplifies all uses or omissions of the articles in Spanish I'm afraid.

Saludos cordiales