1 ) This is impossible to measure. Every single client has a different "demand function", that also varies considerable over time. Who would have thought that toilet paper will ever be sold out prior to COVID? Even if you skyrocket prices (most natural approach), there would be a massive backlash in the media - which in itself hurts the reputation (of the seller, but also the platform).
In a real world scenario. Would the seller have had higher revenue based on a price of £579 or $589? How about toothpaste, what is the (aggregate) elasticity of demand for toothpaste and how about substitutes?
Even the model the platform itself uses to get income is not clear to determine. For instance, revenue maximation is not equal profit maximization. On the other hand, Amazon shows "short" term profits may not matter as long as in the long run your model succeeds.
2 ) Of course, just google pricing mistakes (not systematic - which is difficult to determine in the first place, but obvious) - you will find plenty of news articles where M&S, Amazon, Asda and co had negative publicity or customer / media backlash. In case of Amazon for example, it had a so called Price Parity Agreement. This was under scrutiny for antitrust issue. Nowadays, Amazon "forces" 3rd party seller to abide to a Fair Pricing Policy.
Other platforms like Ebay have similar contractual obligations. Insofar, they have not only an incentive, but direct, contractual rules that aim to help the platform's sellers to get the "best" price.
3 ) While algorithmic trading in finance is indeed a big deal, it cannot be compared to retail selling platforms. It is like comparing a Formula 1 car with a mid-sized family sedan.
- listed products (exchange traded) have well defined supply AND demand data (order books)
- OTC (over the counter) products have usually a multitude of quotes available on platforms like Bloomberg and the underlying products are a lot easier to price. A issuer callable floating rate bond may be hard to price properly, but everyone knows exactly how the cashflows look like, given certain conditions.
- everyone can sell and buy the product within seconds (if needed, even commercially available software allows for 20,000+ orders per second per single connection.
However, google "price discrimination cookies". For example, the [U.S. Department of Transportation (DOT) has approved a new passenger data collection system. This essentially allows airlines and travel agencies to collect personal data, such as marital status, addresses and travel history, to offer a “more agile pricing and more personalised offerings”. It was revealed that the online travel company Orbitz shows MAC users different, primarily more expensive, travel options (e.g. costlier hotel rooms) than windows users.
The book Information Rules offers a great non technical explanation and summary of examples how e-commerce can engage in price setting. This is somewhat dated, but general principles do not change and it shows that price setting was already supported by algos a long time ago (1998 is a long time in computing).