Why is the sentence [2] (fill in "broken down") in the present perfect?
I presume the textbook authors were trying to make the question easier to answer. The choice is somewhat arbitrary. (Choices like these can have subtle effects on the meaning of a sentence in context, but in a textbook like this, there isn't any.)
Is this one natural?
The grammar isn't unnatural. However, “broken down” means something different: it suggests that something with moving parts, usually a mechanical contraption used for transportation, has physically degraded enough that it no longer serves its main purpose.
You could also use “broken”, but that's still stronger than “something wrong”: “broken” suggests that the phone can no longer be used, whereas “something wrong” could be as benign as “the colours on the screen are a bit fuzzy”.
I would fill this gap with the single word “malfunctioned”, or the two words “gone wrong”.
I might also say that the phone “is playing up” (if the issue is currently occurring), or “my phone's been playing up” (if the issue is not currently occurring, but that I expect it to reoccur shortly), but Wiktionary says this usage is chiefly British, so it may not be widely understood.
Can we write it in simple present tense?
Yup!
But, a couple of students are wondering, why can't we fill in "something wrong" here.
If someone “has something wrong”, it means they have a mistaken understanding:
Alice: "I was talking to Carol about my new kite, and she said you might want to fly it with me."
Bob: "No, Carol has that wrong. I like the birds, not the other sort."
The textbook's sentence provides enough context to make it clear this is not what is meant, but it's still confusing to process. We would normally write:
My mobile phone has something wrong with it, so I'll get it fixed.
The transitive and intransitive forms of an expression often have different meanings in English, so it's important not to mix them up.