Explain PSN/SEN/PS Store to me?

Sony Entertainment Network was the collective name for all of Sony's digital media services, The Playstation Network was part of that, relating to PlayStation consoles. However earlier in the year, the 'SEN' name was retired and replaced by 'PlayStation Network', into which the SEN's music and video services were absorbed. The Playstation Store is the storefront for that service. So in essence they are all slightly different parts of the exact same thing.

You can see your whole list if you log into the SEN/PSN website, but sadly no, there is no way to sort the list, not by platform, nor by anything really. It's really annoying, since on PSVita's own download list it shows everything I own, including PS3/PS4 crossbuy titles, PS3/PS4 DLC that came with those, stuff I can't even use (and especially not on the Vita itself), with no way of just getting a list of actual Vita games.

Im going to say no, but if you take into account that your downloading them to a memory card, then maybe it will work.

on ps4 there is an app called library but that only shows your ps4 games.

It's a store you open via an app. You select icons within the app and buy stuff with a credit card or a psn card. At the very bottom and sometimes top of the main menu is an option called "downloads". You click on that and get a chronological order of things you downloaded, not just purchased.