So the answer also states 'we are searching the cat'. How would you go about writing a statement that is unambiguous about whether you are 'searching the cat' or 'searching for the cat'?
"We are searching the cat" = Wir durchsuchen die Katze (e.g. the cat ate an object and we need to look inside it)
"We are searching for the cat" = Wir suchen die Katze (e.g. the cat is missing and we want to find it)
"We are searching for (on behalf of) the cat" = Wir suchen für die Katze (e.g. the cat can't find something and asked us to look for it)
Can you also say "Wir suchen fur die Katze" ? How would you say "we are searching the house for the cat" (not in the house)
No, you can't. You can say 'wir suchen etwas für die Katze', but then you are searching something for your cat, not your cat itself.
No, it wouldn't be the same :) because it should be: "Wir suchen nach der Katze"
Yes, both "suchen etw (accusative)" and "suchen nach etw (dative)" work.
How can suchen mean both "seek" and "find" at the same time? I imagine this can easily lead to a misunderstanding. Does the context make the difference in the interpretation of the exact meaning for the word "suchen"?
"We are searching the cat" means that we're looking for something that is hidden on or inside the cat.
"We are searching for the cat" means that what we're looking for is, in fact, the cat.
How does one say "we search the cat"? (in the same sense as "we search the bag")