I am looking for an application that would allow me to read (summaries of) all kinds of inboxes and social media such as Gmail, Facebook, Disqus, StackExchange, Twitter etc.
Is there such a thing?
You can use gmail. And also let it fetch your pop mail messages. All of the other things you describe (at least fb, stackexchange, etc) have the option to send email alerts...
I use Gnus (in emacs) which has back-ends for many of those (I don't use them all, so I'm not certain about the ones I don't use). Although Gnus was originally developed as an NNTP client, its architecture makes it easy to substitute a back-end to talk to any message store, most mail protocols are supported as well as direct access to local mail files. And back-ends have even been developed for some systems that don't bear much resemblance to messages (e.g. man pages).
Personally, I use IMAP for access to GMail, my own mail system, and a few other places. It also does POP for mail (I just don't use POP any more). You can also set it up for reading RSS feeds, I know. I've seen mention of most of those others on the mailing list, but since they don't apply to me I didn't commit them to long term storage.
I looked at my currently running Gnus, which shows 19 different back-ends currently loaded. There may be more that have not been loaded, since I haven't referenced them. These are their names (some of which are obvious and many of which are very specialized and may not be at all meaningful), the "nn" at the front of each is there for historical (some might say hysterical) reasons, ignore it for all but "nntp":