'For what are you looking' is the correct grammar in English as sentences should not end in a preposition.... therefore it should be accepted.
Yes, in English it's not grammatically correct to end a sentence with a preposition (assuming that's still a rule..). However, as a native (US) English speaker, I don't think I've EVER heard anyone ask, "For what are you looking?"
It is VERY common in everyday speech to find a preposition ending a sentence, rule or no rule. I think the course is probably trying to find the balance between "absolutely grammatically correct" and "real world" English in the exercises, which seems like a good idea to me. Others may have different opinions.
Not sure if you are right or not but it only supports my claim that Czech translated directly to English sounds vaguely like a medieval knight!
You are correct! I am a native (US) English speaker and former English teacher. I actually tried "For what are you searching." This was also counted wrong. While it is not commonly used, it technically should still be counted correct as it is proper grammar.
According to Google ngrams (links below), "For what are you looking" and "For what are you searching" are used very rarely, despite the "don't end a sentence with a preposition" rule. Since ngrams searches written works, it would seem that the rule has fallen by the wayside both in speech and in writing.