With the digital upgrade to the Taken King, you will be entitled to play Destiny without the disc, but will still be required to download the complete copy.
Games run a little bit differently when you install them, compared to downloading them. Best thing to keep in mind: Games are made available online so you have easier access to them, not so that you can buy a cheap version of a retail game you own, and give the physical copy to a friend1. Games install from a disc so that the console has the files it regularly needs to access closer to home. This does not mean all of the files it needs.
When you install the game, it installs in a form that tells the console "when this game runs, check that this specific disc is in the drive, and use it as a reference point for other data". As far as the console sees, that install comes from a disc. It does not even check to see if you have a license to play the game online, as it assumes that you have the disc available to you.
While the online version would simply be adding to the original install, the two methods of playing Destiny are intentionally separate. For this reason, the online version does not "integrity check" the offline version, and vice versa. You could download the additional files from the download version of the game, except there is no way to check what files you have, and what files you do not.
The only real solution you have, here, is to simply download the entire original game from the PlayStation store2.
1Don't get me wrong, this is a perfectly good reason to do just that, but the developers have not set up their system to support this practice.
2My research to confirm the difference between an install and a download on the PlayStation 4 also reported that the PlayStation Store, itself, is often responsible for slow downloads. Given your reports that your internet is normally significantly better, I suspect the bandaid fix of "take the console to a friends house, who has better internet" would not be a good solution.
For the discs part - no. The disc is used to validate that you own the game. Even if you install the game off the disk, the console will expect to see the disc. The PS4 uses the disc to basically say "Hey! You really do own this game!" If this wasn't the case, you could essentially buy one disc and share it among many people and they would all have the ability to play the game if they install it to their PS4.
For your downloaded games - yes. You will just have to download the game onto both consoles. As long as you are using the account you purchased the game with, it should work. Digital purchases are linked to your account.
Best Answer
No, this is not possible.
Games must have been purchased/downloaded from the PS Store in order to be installed via the library. Physical versions aren't tied to your account in the same way as digital, otherwise I could just sell all of my physical games immediately and redownload them digitally, allowing me to play for practically no cost.
You sold the physical game and therefore no longer own a copy. You must buy a new one. This is unlike digital games, where you obviously cannot sell away your copy, so you are free to redownload it whenever - you still "own" it (or rather, you're purchasing a license to use it, following Sony's Terms and Conditions).
Assuming you've actually deleted all data associated with the physical copy, you shouldn't run into any issues installing/playing the digital version.