I purchased a month of EA Access for PS4 and I started the download of Need for Speed Payback and Battlefield 4. Is all of the DLC that comes with those games via EA Access going to be installed automatically or I need to download them separately?
Unfortunately no, you will have to install them separately unless you've bought the unlimited or the ultimate edition of the game, in this case it comes with the edition of the game.
Further more, if you downloaded the standard edition of the game, DLC's will come with a price as in The Sims 4 for example, DLC's are at a 10% discount - so watch out for this.
Always look for the complete package, deluxe, ultimate or unlimited.