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Actually, there is a difference. Weit is not commonly used for distances (i.e. in the sense of "far") without being accompanied by "entfernt" or "weg". weit standing alone can be used for clothing, meaning a wide fit, or for a wide landscape (i.e. in the sense of "wide", a close relative). Er wohnt weit weg von uns Er wohnt weit entfernt von uns. ...
The following two sentences are correct. They both express a geographical distance. Er wohnt weit entfernt von mir. Er wohnt weit weg von mir. In your first sentence, Er wohnt weit von mir, I am missing something. It's not wrong, but I would like more context. It could be a geographical distance, but it could be also a poetic expression of an ...
Your choice. There is no difference, both expressions mean the same and are used in exactly the same context. Very much like in English on account of s.o. on someone's account
Auf Kosten anderer is a little more "sophisticated".
Actually there is no difference. I think most commonly you would use: Er wohnt weit weg von mir.
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