I am not sure to what extend board manufacturers do bare board testing.
You could always ask, but generally its a quick visual check at each stage of manufacture. The guys (and girls) on the production line are usually pretty damned good, but they are not infallible. With a ten layer board, the chances of a few duds sneaking through on bad days increases. Only way to be completely sure is to pay the extra for a fully automated (no bad days!) probed test of the finished boards and see if your yield increases.
Its a real pain, but its the only way to be absolutely certain.
Another thing to enquire into are the design rules for the PCB. Hair thin tracks and ultra fine clearances are going to be more problematic than bold tracks and larger clearances. Your CAD design people should be able to tell you. Drawing up a PCB that is very difficult to manufacture is not going to do you any favours.
As far as chips go, buy direct from the chip manufacturer, and make sure you have an honest purchasing officer. Counterfeit "Dragon/Panda brand" may be cheap, and come with a really nice Christmas present for your purchasing officer, but it spells doom to your business. He may feel he is doing his very best by saving your company money, but it may not be a bargain. Enquire into this...
As far as assembly and soldering goes, your methods could range from truly awful, to superb. No way of really knowing until you eliminate other possibilities.
I guess it all finally distilles down to professional know how, work ethic, and integrity throughout the entire chain.
One final thought.
ESD damage during manufacture due to poor methods of component handling is not unknown. Do staff have adequate training and the equipment to do the job ?