Yes! I have seen the stuff Yelp is doing. Reviews are definitely integral to Yelp so they must eliminate those fabricated reviews to maintain their reputation otherwise no one would be using their website.
1) You are correct, a large number of the fake reviews are actually a result of agencies that offer their services to manufacture high star reviews to increase their product ranking. Now, these people use bots (or an army of trained monkeys) for most of their review entries and Amazon has in the past detected them. However in the past year or so I've noticed a lot of fake reviews still lingering in Amazon and that is a shame as a lot of the time they mislead consumers. This is why I created the site as I have fallen victim to it too. Just take a look at the health supplements section of Amazon and look at the amount of dishonest marketing some companies are doing.
For case 2), at the present moment Fakespot would be unable to distinguish such a scenario where a reputable reviewer was paid by a company because that would be indeed a "real" review written by a real person. However, a large chunk of the analysis algorithm does look at the language and if the review text is extremely positive and not very detailed to relevant to the product, it will raise a flag.
3) For false positives, the machine learning implementation records votes and modifies itself to eliminate them.