Is it permissible to write the name "Ar-Rahman" with an alif (الرحمان), whether in the name "Abd Al-Rahman" or in phrases like "May Ar-Rahman bless you," and is it valid to use it outside the context of the Quran as a personal name?

Writing "الرحمن" (the Most Merciful) with the alif in the Quran is an error, because the orthography (rasm) of the Quran is a followed tradition that must not be violated. As for adding the alif in texts other than the Quran, it is a lesser mistake, but it is still an orthographical error according to the rules of Arabic dictation (imla'), which stipulate the omission of the alif due to frequent usage.