Do All The Daedric Artifacts have to be collected with the same character

It's connected to Pieces of the Past, which unlocks when you're level 20. Once you turn level 20, a courier will inform you of a museum opening up in Dawnstar. There, you'll run into a guy named Silus Vesuius who begins the quest.

During the quest:

He'll task you with getting all three parts of Mehrunes’ Razor. Once you do that, he'll join you when going to the shrine, and will unlock it.

Just as an additional aside (and major spoiler):

At the end of the quest, you'll have a choice between killing Silus or letting him flee. You must kill Silus in order to get the Daedric artifact.

Once after that, two Dremora will appear at the shrine. Kill them and take the keys to get in the shrine. Inside you'll find 2 more Dremoras and there are so much loot.

But hey! Don't go away!

Wait 10 days and the Dremoras will respawn and the shrine will be reset. You can kill the Dremora, and then wait for 10 days to get more, leaving a good method of getting unlimited daedra hearts.

To answer the question in your title, PS3Trophies list five of them as missable. Here is the list:

• Mehrunes’ Razor
  Sometime in the game you’ll get a message from a courier to visit the museum in Dawnstar – visit it. Visit it even if you don’t and start the “Pieces of the Past” quest to get your hands on Mehrunes’ Razor, which will send you to three places in Skyrim to get the three pieces of the sword.? You must kill Silus at the end to get the artefact.
• Savior’s Hide
  In Falkreath at the Falkreath Barracks (east of town), downstairs in the Falkreath jail you’ll find a chap called Sinding who will give you his cursed ring and thus starts the quest, “Ill Met by Moonlight.” Again, like the rest, you must choose to kill or risk not getting the reward. You cannot have the Ring of Hircine and Savior's Hide. You must kill Sinding instead of helping him. Thanks to jamby for the info. It is possible to get both Savior's Hide and the Ring of Hircine and have them both count. You will have to kill the hunters and then go back and kill Sinding. Thanks to Rawra for confirming.
• Masque of Clavicus Vile
  The first time you enter Falkreath through the front gate the guard will mention something about the blacksmith’s dog going missing. Speak to him and then go after the dog. When you meet the dog you’ll initiate the “A Daedra’s Best Friend” quest and head off to Haemar’s Shame at the south of the large mountain with High Hrothgar on it – east of Helgen – note: you have to follow the dog but he has terrible path finding, so you could just meet him there.
  You’ll then set off to retrieve The Rueful Axe from Rimrock Burrow and return it. NOTE: Do NOT kill Barbas with the axe when you return, give it back to Clavicus and get the Daedric masque.
• Skull of Corruption
  If you head to the Windpeak Inn in Dawnstar you’ll see a bit of a situation arise with locals speaking to Erandur regarding some night terrors they’ve been having. Speak to Erandur and start the “Waking Nightmare” quest. It’s a simple follow him, go here, go there trophy (note: the Dreamstride book is on the balcony area in the far corner) but at the end when you get the chance to kill Erandur whilst he’s performing the ritual, kill him and you’ll get the Skull of Corruption. If you don’t, he destroys it.
• Ring of Namira
  In Markarth, in the Understone Keep you’ll find a priest called Verulus who wants you to head into the Hall of the Dead to investigate what’s going on. The quest starts off as a miscellaneous objective and will soon turn into the quest, “The Taste of Death” once you’ve spoken to and agreed to help Eola in clearing Reachcliff Cave. There is a potential that you could say no to helping Eola… don’t! In order to get the ring you have to kill Verulus as he lies on the table and then eat his flesh.

Now, for what you put in the body, jwaddell just answered and I don't think I can add more. You can still try what Daniel suggests in his comment on your question, which seems to be the only way to get the 15 items if you're not on a PC...!