Under what conditions is using some other brand's logos and pictures on a commercial website allowed? Could I use, for example, a picture of a Dell computer (taken from their website) of a given model to let users know I have it for sale?
As a legal matter, somebody (likely Dell) owns the copyright to their photos and copying them without permission is a crime, and probably a violation of their web site terms of service. An aftermarket seller could contact dell and inquire, but it would usually be faster to clean up and take a picture of the one you wanted to sell or get permission from someone else who has taken a picture of the same model (but if the latter, be clear in your advertisement that it is the same model pictured, not the exact item pictured).
There may also be trademarks in the photographs or in the listing person's text. Trademark infringement is about consumer confusion. So anyone selling a product and using those to identify the product would want to make sure that they cannot be construed to mislead anyone about the origin of the product--that would be trademark infringement. So if you say it is a Dell computer model so-and-so but you have swapped out the motherboard or it's actually an ASUS, or if it says Intel Inside but it's actually an AMD chip, that could be trademark infringement.