I can't speak to marshalls specifically since i haven't seen their paperwork myself, but I wouldn't be surprised if we can specify storage conditions, it's not something that's unique to high end places. Still no guarantee that anyone does it, but it'll be right most of the time. A lot of discount retailers price returns into what they pay us, they just take 0.5% (or whatever is appropriate for the category) off the top and we don't don't have to handle it ourselves unless they go bankrupt or the product is flipped online and the customer reaches out to us instead of Amazon.
As for lot codes, we can (and do) put them on packaging but it isn't really feasible to put them on the product itself. We can track product where someone has an issue with something they bought 3 months ago from a specific store and can in most situations identify 1 or 2 possible production lots that it came from.
Some of our products have a lifetime warranty, Amazon surely aren't going to honor that for us on a gray market purchase.
Either you've got to mandate that all warranties apply no matter the supply chain (and presumably that'd have to be done in international law) or you'll have manufacturers/brands who can set their own warranty conditions to only apply when sold through authorized channels. Amazon can't really be complicit in passing off items as "New" that don't have a manufacturers warranty. If someone falls to their death because they bought a piece of climbing equipment from a lot that the manufacturer had recalled then Amazon don't want to be the one holding the ball - who can blame them.