If you want a short form of your own name, you could also spell it 'Mani'.
'Manny' is the most English spelling of this name, but (to my eye, at least) 'Mani' is more suggestive of an Arabic nickname.
I wouldn't choose 'Manie', if I were you, because we would pronounce it 'Main-y'. This is a little bit strange, and suggests 'mania', which is not a good thing.
'Manne' is not a normal English spelling, and people might think it's pronounced 'man' - like the name 'Anne' - or maybe 'mann-uh'. This spelling might confuse people.
If I had to choose a short form of 'Abdurrahman', based on the last syllable, I'd go for 'Mani'.
I hope that helps.