Good investigative work, FlyingMan and MagicTrick. You could always write back to them saying that they dont need to worry about processing your paperwork as its already being arranged through another 3rd party (e.g. your embassy). See what they say...
I think its amusing they offer such a great package ($12,000 a month - without even meeting you!) and then cant afford to pay your travel/processing cost (!) and ask you to pay the $350.00!
yeah that adds up.
But seriously the biggest concern is the consistent request for family contacts etc... it could be a little more sinister than it appears.