Legality of collecting/distributing a dataset of screenshots of websites
Given the state of copyright law, this sounds like a minefield. There are people trying to sue Google, just for indexing sites. 

It isn't so much that you want to do research on these "images", which is likely ok, but I think that distributing them will be almost impossible. Many sites won't want to give you permission and will complain (via lawyers) if you haven't obtained permission. 

The general law about copyright is that people own what they produce, whether they register a copyright or not. If there isn't a clear license offered such as a Creative Common license, then the owner retains all rights to the work unless they explicitly put their creation into the public domain. Most of the sites you will probably want to visit likely will NOT have given up rights. The deal is supposed to be time limited, but that idea has been nearly eliminated in the law. 

Places like wikipedia have clear statements of licensing and rights transfer. But individual users at reddit, twitter, etc. likely still have rights unless they have yielded them (as part of signing up) to the hosting site. I don't participate much in social media so haven't investigated these user agreements. 

Attribution isn't enough. You need permission. Once you have that, you need to provide attribution. 

However, if you want to "process" images, gleaning characteristics of the sites (but not text and images...) you can probably do it without permission. 

Also note that copyright law is in flux and varies greatly around the world. The EU is currently trying to work out very restrictive laws, for example. 

The core problem is that IP owners often have a lot of political power. Disney, for example will declare war on you if you try to compromise their rights to Mickey Mouse. Common understandings that once permitted Voltaire to make a living from his writings have now expanded to unlimited rights to lock up the culture forever by large commercial firms. 

Another problem is that many formerly lucrative publishing strategies have been money losers lately because of the internet. The NYTimes, for example, if it wants to continue in business needs to achieve income somehow. They can't do it printing on paper anymore, so their web sites have value to them and they are fearful (maybe overly fearful) that re-publishing, or even indexing, their material will cost them enough that they can't sustain their business model anymore. 

It is even worse for sites that show advertising. You not only have issues with the site itself, but all of the images shown by the advertisers are, themselves, likely copyright protected with teams of lawyers ready and willing to go after you. 

That said, you might be able to do your research on a wide range of sites/images without formal permission, depending on what it is, but then only publish a smaller set for which it is clear that you have a right to do so. 

Note that there are a few exceptions to strict copyright law. These fall into the realm of formal criticism, and parody. There is a notion of fair use that is being eroded, so can hardly be depended on anymore. 

If you are part of an institution, such as a University, then the institution probably has an Office of Research that can give you explicit advice for your locale. That office probably has access to lawyers that can provide explicit legal advice. Tread cautiously.