"I am looking for my cell phone."
Otherwise it would be "I look for my cell phone;" 在 here basically does what '[present of to be] + -ing' does, in English.
But I look for my cell phone is not correct english, and i am looking does not indicate a continuous form, but is rather just an expression
"I look for my cell phone" is a proper sentance. For example if you asked me "what do you do?" I could reply "i look for my cell phone." Most times we assume "what do you do?" to mean "what is your occupation?" but its not improper to mean it as "what are you doing?" Or i could be joking and say "i look for my cell phone. Its my occupation!" Its the differance between the present simple tense and the present continuous tense. Both are correct.
I dont get it. For example the sentence, "I look for a job" clearly sounds wrong to me, it should be "I am looking for a job". Even if my current occupation is making dinner. Then there is your example, but it has a completely different meaning, not searching for something, but looking after something.
You may be asked "What do you do when you lose your cellphone?" and you could certainly respond "I look for my cellphone."
Because 在 as well indicates a state or something in progress like the English "ing-suffix".
You do not need 在 here. In fact, (depending on the context which we don't have of course) it might sound awkward. In English, we almost always use the -ing suffix to indicate an ongoing action or that someone is in the process of an action, but in Chinese （正）在 is what emphasizes this idea. (正）在 can be omitted because if you simply say 我找我的手机，the listener knows you mean "I am looking for my cell phone," provided there are no other contextual factors which would suggest otherwise. This would not sound weird to a Chinese person like "I look for my cell phone" might sound unusual to an English speaker, and they are both grammatically correct.
In general Chinese is in many ways very simple compared to most Western languages, and certain indicators and words can often be omitted if you know the context of a sentence. I hope this helps!
Why it is wrong without 「在」？Here were 你找谁，你找什么 and other － all without「在」，all translated in continious and it were correct answers.
I may be out of line saying this, but i know in Japanese, Spanish, Portuguese, and Korean you can usually omit such pronouns as I, especially if the subject is understood, and I don't often hear 我 used by native speakers more than like once or twice in a conversation
As I understand it, 是 is used for linking nouns, which in English is like using the verb "to be", as in "I am an X" (where X is a noun, not a verb). In this exercise the "am" is just part of the English present continuous tense which can be indicated in Mandarin by using 在 before the verb (but not 是)