I'm not a native English speaker... is "We search a dog" totally wrong?
It is not totally wrong but it means something different. "We search a dog" could mean that you are looking or searching for something on a dog. For example "We search a dog for fleas"
Yes, it's wrong. Use "search for" or "seek." Seek is a better choice for cauta if you don't want to deal with the preposition.
I search for the dog. I seek the dog.
@DennisVanBeek - It is wrong in a literary language, but an English speaker would understand what you say.
I think you understand the next line. De zin klopt niet helemaal. To search for is meer we zijn op zoek naar.