Yes, αυτή and αυτοί are homphones, which means that they produce the exact same sound. We can't solve the listening exercise problem ourselves, so we just keep on reporting them and hoping that they'll get fixed in the near future. :/ Thank you for your comment though. ^.^
That was a really old rule that was once applied to English through Latin grammar. It's prescriptivism that has essentially fallen out of use. It's more important to abide by what's considered contemporary natural usage rather than go back to what was being taught centuries ago.
In English there are fixed forms for word order. What you suggest is not one of them so is not possible as a translation here.
Also, please read the comment by here concerning the use of the preposition at the end of the sentence which is correct and not an error as was once believed .
I'm a little amused that you worry about the rule about "don't end a sentence with a preposition" but not the one about "don't split an infinitive" ("to not end").
Both of them are prescriptivist rules that many native speakers do not use naturally in their language -- and so the result often sounds unnatural to them, and worse English than if they speak normally.