The bulk of the game will still be available to you after the servers go down; there is essentially no impact to the campaign. All the stuff listed above is essentially asynchronous multiplayer stuff that sits on top of the main game. Treasure hunts are fun little activities where you have to search for a hidden treasure chest in a small area using only a photo taken by another player as a hint. Challenges are essentially the challenge missions scattered throughout the world, but you're competing against other people's results instead of the arbitrary bronze/silver/gold standards in the single-player challenges. Photo ghosts are basically just ghost versions of yourself scattered throughout the world; when you find one, you can see the photo someone took while standing at that spot.
Interacting with all of these gets you Dusty Tokens, which then unlock various alternate costumes and some special talismans. Treasure hunts also unlock various gestures and furniture items you can use to decorate Kat's home after a certain chapter. None of these are required for the campaign and nearly all of them are purely decorative. The talismans are the only items that affect gameplay, and while the 6000-token talisman is great for exploring (unlimited gravity shifting but lower attack damage), you definitely don't need it to complete the game or any of the sidequests.
The people who are farming Dusty tokens are completionists who want all the costumes and items before they're no longer obtainable. If you don't really care, feel free to play Gravity Rush 2 on your own timeline.
