Ps4 – Is it true that you need to be always online for a game once you download a DLC for the PS4

Ok, I don't have a PS4 so I can't test this out, but according to this article yes

Summary:

Shareplay only works with two players for a maximum of 60 minutes at the moment, but plans to develop and extend

while it has some constraints - such as PlayStation Plus requirements, 60 minute session lengths, and is limited to two players at launch - it looks to allow games to be shared with another user quickly and easily.

One player (the host) creates a private party and invites the user, then you share (I assume via the share button on the controller), the host continues to play until the invited player joins, from there the host chooses to either let the invited player take control or alternatively play together. Thus a local multi-player game will work online

When you begin Share Play, a private party must be created and a user on your friends list invited. The host can continue to play the game until the recipient accepts, where by default, the recipient can watch a non-interactive mirror of the host's game.

From there, the Host has two more options with the Share menu. One is handing over control to the user, while the other is to play the game together, so that a local multiplayer game can be played online.

The answer is simple: No you dont need PS+ to download games or dlc's. Entering keys you bought on any site is the same like buying it in the PlayStation Store. However you'll need a subscription (most of the time) if you want to use the Multiplayer features of such games.