Does Persona 5 Royal have all of Persona 5’s DLC

In order to prevent spoilers, ATLUS blocked the PS4's native share functionality for the entirety of the game. This includes saving screenshots and videos. Thus, this isn't an issue with your PS4, it's a restriction placed on the game itself.

For reference, see this article:

Atlus confirmed to Polygon that because Persona 5 is such a story-based game, the company made the decision to prohibit players from sharing content.

“This being a Japanese title with solely a single-playthrough story means Japan is very wary about it,” an Atlus representative said. “Sharing is currently blocked through the native PS4 UI.”

Note: as mentioned by @MadMAxJr in the comments, this includes screenshots typically taken when earning trophies. Thus, there will not be any screenshots taken when earning trophies in this game.

According to the wiki, Persona 5's New Game+ carries over the following:

• Human parameters, the five status
• Persona compendium
• Equipment, including teammates'
• Money
• Skill cards
• Items that unlock persona
• Items received from maxing co-ops
• Xmas present
• Play time
• Fishing rod
• Valentine chocolate, or The pity chocolate from the cafe owner

This means that the following things are not carried over into New Game+:

• Name
• Game difficulty (Player can re-decide)
• MC's and teammates' level
• Consumable items
• Co-op ranks
• MC's personas
• Tutorial and Story cut scene history
• Materials for crafting
• Presents (for Co-op)
• Books

Thus, to address the part about player level, this will be reset, but you can potentially buy back your end-game level personas as soon as you start, so even though your player level will be reset, you'll have extremely powerful personas.

That said, a good tip for preparing for New Game+ would be to focus on getting your Social stats as high as possible, since those do carry over. The less time spent raising Charm or Guts will be more time spent increasing your Social Links!