Many times, yes. "MSRP" stands for "Manufacturer's Suggested Retail Price".
Now, not all products come with MSRPs, and unless there is a contract between producer and retailer on a pricing floor, the retailer is free to set the price of a product as high or low as they want.
But, there is a key difference here. Let's use your supermarket as our example. A supermarket sells both Rao's Pasta sauce (7.99) and MarketBrand Pasta Sauce (2.99). The difference is that the store has paid Rao's 2 dollars to purchase their sauce which they then mark up to 7.99, and had their store band produced for them at a purchase price of 1.25. Both manufacturers have been compensated for their products.
Instead, what Amazon does is take a shipment from Rao's with the promise that they will be compensated 2 dollars for each bottle that is sold (which requires Raos to tie up inventory/money) and then undercuts their price with a product that they produce themselves and list on their page much more prominently, after using Rao's product previously to demonstrate market demand for that type of good in their store.
Depends. Most of the merchant decides. However for "premium" brands the manufacture sets the price and won't let you go below it.
Weber grills rarely go on sale or clearance because you need permission to do so. Menards near me often has 11% rebate on everything sales - they do not sell Weber grills. Home Depot matches Menards 11% rebate on everything, except things Weber grills where they are not allowed to discount.
I know of one case where a golf store threw in a free hat with the full price purchase of an expensive putter. The customer was an undercover agent for the putter company and the store no longer got more product from that company.
I know there are a lot more examples, but I don't know them.
Vendors will do rebates so they do have some control over price. Like new cars when a dealers sells at their cost, there is a large rebate the customer doesn't see.
Yes merchants/retailers almost always decide prices. In fact some variant of "pricing is at the sole discretion of the retailer" is a standard email signature and PPT/Doc footnotes of many sales people in most manufacturers.
