My experience is not exactly what you describe, distributors are there to deal with the small to medium customers that the factory does not want to directly service. For instance TI doesn't care much if you want to buy 1000 MSP430 chips, but digi-key might, or maybe even Arrow. So unless your volumes are large, or the factory is small you will not be buying direct from the factory, that choice has already been made for you :) If you were able to buy parts direct from the factory I would not expect a problem with counterfeit goods, just as I would not expect one from say an Arrow, Avnet, Future, Digi, Mouser, etc. Those are examples of large and franchised, or authorized distributors for various components. (I'm using factory the way buyers and distributors use it to mean the manufacturer such as a TI, or Samtec for example).
Someone may know otherwise but I've never had a distributor test parts for me, although I have had them program and test things like flash or MCUs. I suppose if I complained they would be able to charge me for such a service but I don't think that is routine. It's the factory's job to test their components before they ship them. As such if you have an issue the disti will point you back to the factory for such things. If you are a good customer, and you're angry, they may take the parts back directly and work it out with the factory themselves, but probably not.
I've never seen any online reports about distributors, they may exist. I tend to stick to people I know and I've worked with in the past such as the names I mention above. That said there are lots of smaller distributors around the world, and my only word of advice is if you suddenly find yourself getting a great deal compared to the other quotes you've received perhaps it's too good to be true :) Another thing is I prefer to try to foster a good relationship with the distributors I do use.
For the last question I'll break it in two. If you bought parts from a franchised or authorized distributor, and they turned out to be fake, that disti would have a lot of explaining to do. In that case I would expect and demand a full refund, and might even demand compensation for damages if it was serious enough. If you got them from another source, say a broker, an unlicensed distributor, another CM or god forbid ebay. You're pretty much screwed.
One of the things I've noticed lately is that large CMs are selling parts they're buying at a volume discount out the back door through various brokers or agents. In fact there seem to be whole companies that have sprung up just to support this. I personally think that's a dangerous game to play if you're worried about quality and genuine product, but it does seem to be popular.
Now what if they don't work or half work after you've assembled them? Assuming you have bought authentic parts from an authorized disti or the factory then the factory should engage you to help do failure analysis. Now your mileage with this will vary depending on how good of a customer you are, and as a small guy you may spend some time talking to the disti's support team to make sure it really is a failed part. Once it's done most people will engage you because it's in their best interest to catch and understand any field failure.
A few notes on that process, one it takes forever, you may be better off analyzing it yourself at a lab and sending them results and samples of the failure. Two there can be a lot of finger pointing, the CM says it's the factory, the factory says it's the CM, or they both say it's your design or spec. So you have to stay on it to find the answer, and of course be open to the possibility that it is in fact your fault. Finally the level of support you get here will vary based on what factory/manufacturer you are dealing with, your relationship and volumes with them, and indeed the region you are in. So while I have seen for example linear tech jump at every problem I've ever uncovered and sent their way, I would hardly expect that kind of service for the $0.10 regulator I'm buying from an overseas off-brand.