Translation:Are you searching for a big dog? It is over there.
It should be accepted, but "a big dog" is as likely as "the big dog".
Scenario: You are going for a walk in a park. You spot a big dog. You keep going. You notice some people looking around, maybe like there are looking for something. You ask them: "Are you (by any chance) looking for a big dog? (I just saw one.) It's over there!"
Google suggests that difference between look/search is how much effort is used for the activity. In this case, one usually puts much effort to find their(?) dog, so "search" sounds better word.
Generally, "etsiä" can mean both. If you want another word, "katsella" could be used (katsoa=to look), but I would use that only when you are wandering in a shop to find new clothes, so you are not searching a specific piece of clothing
I actually came here to say that "looking for a dog" is more idiomatic that "searching for a dog". All of these examples with "search for" sound like Finglish to my ears. :)
Perhaps it's less about the effort, and more about the thing being sought: I search for meaning, for the right word, for life purpose, for a feeling, but I look for a lost sock, for a building, or for the butter.
Adding -ko/-kö is a question thing, if you don't have a question word (why=miksi, when=milloin, how=miten, etc), then you add the ending.
You are searching! = Sinä etsit!
Are you searching? = Etsitkö sinä?
When are you searching? = Milloin sinä etsit?
When to choose -ko/-kö is indeed based on vowel harmony (I don't know Hungarian). Basically any one Finnish word can only contain either: a/o/u or ä/ö/y, where i/e can be combined with either. We do have also å in our alphabet, but that is virtually never used.
Base word here is "etsiä", so you need to choose -kö.
For completeness sake, this vowel harmony does not apply for words in foreign origin (like "olympia" (very hard to pronounce correctly by finns)) or combined words (my favorite is "hääyöaie", which is combination of "hää" = wedding, "yö" = night, and "aie" = intent, not the most usable word thought :) )