Yes, there actually is a list with all contents you have ever "purchased" over PSN (for free content, "purchase" means the first time you downloaded it). That list includes content from the PS3 or PS Vita (possibly also PSP), regardless of which system you are currently using to view the list.
Unfortunately, the list isn't very user-friendly; the more you purchased, the more difficult it is to find what you're looking for. It is sorted by date of purchase and it is currently not possible to change the sorting order.
Here's how to view the list (I have German localization, so the options' names are likely not accurate):
PS3:
Launch the store app, then go down to the third-to-last option, called Show downloads.
PS Vita:
Launch the store app, then touch the ... button on the bottom right corner of the screen; touch the second option from the top, called Download List.
PSP:
(I do not own a PSP, so feel free to edit this point)
Any and all content featured on that list may be downloaded again for free. The only limitation is that you can't download PS3 content to the PS Vita (possibly also the PSP), and that PSP content can only be downloaded onto a PS Vita, if that content is also available in the PS Vita store (you can still download the content over a PS3 and transfer it to the Vita, although there is no guarantee that the transfer will actually succeed)
As TZHX already wrote in a comment: The PS4 is not backwards compatible, meaning you can only play Playstation 4 games on it. The only streaming capability it has is that you can stream PS4 games to a PS Vita. In the future, it will support streaming games of the PS3 and older platforms online via Playstation Now with a rental system. Only two days ago it went into open beta.
When you open the PS store on the PS4, only PS4 titles appear, so you cannot download PS1, PS2 or PS3 games at all. The only PS3 games that appear are PS Plus titles that you can claim for later download on a PS3, but cannot download on the PS4 itself.
The PS1 games available for download in the Playstation Store can only be played on PS3, Playstation Portable (PSP) and PS Vita, while the PS2 games from the store can only be played on a PS3.
Not all PS1 games are available to download on Vita, and some of them are region specific. Some also work when they're on the Vita but cannot be downloaded directly from the store.
When you access the store on console or at store.sonyentertainmentnetwork.com you can see which platforms are supported by each game.
Best Answer
If you start the download from your console it will only show the progress from your console.
The only way to view the download progress from the PlayStation website is to push the download from there. Once you have selected to push a download from there the web site will display if the software is waiting to download or in the process of downloading.