In this sentence, the i sound in 'bayt-i-ha' is correct, unlike in 'bayt-i-hu' which's supposed to be pronounced as 'baytuhu'. That's because 'baytuhu' is at the beginning of the sentence (the nominative case), and 'bayt-i-ha' is preceded by مِنْ 'min' which's one of many Arabic prepositions that require the word that follows to have the i sound, which's كسرة kasra. (the genitive case).
Check the case lesson for more information.
"his house is next to her house" just sounds awful, because of the repetition of "house". I think that, to make the English idiomatic, the answer should be "his house is next to hers". And this is regardless of whether "his house is next to her house" and "his house is next to hers" have different Arabic translations: these two seem, in English, to have the same meaning, but the first sounds much clumsier than the second.
SyedOmar, languages can never be translated completely word for word. Evidently, Arabic says "close from" not "close to" as in English. Russian also says "close from" (близко от), and French says "close of" (près de). I expect other people can give other different examples from languages I don't know. We can't demand that they all follow the same pattern!