Can I install and use all the DLC on a already started game?

Which DLC have you gotten?

They should automatically appear for the game you've already started. Particularly the addons such as Leliana's Song (which are playable from the main menu and not from your game itself). The addons such as Warden's Peak or Return to Ostagar that provide further quests should be available after you've installed them from your current game

You do not need to restart your playthrough to take advantage of new DLC unless you're at some point-of-no-return (endgame) and want to access one of the DLCs that add a map travel option after the game prevents you from traveling anymore.

Golems of Amgarrak, Leliana's Song, and Witch Hunt are separate campaigns. To access them, you will need to start a new game from the main menu, select that DLC campaign, and select a saved game to import your character information (if the DLC requires your character information / decision history at all). Items that carry over to your main campaign are based on achievements unlocked while playing those campaigns, and those items typically appear in your character's inventory the next time you load any saved game from your main campaign or start a new game.

The Stone Prisoner, Warden's Keep and Return to Ostagar will appear in any existing game as a map travel option, and some in-game element - like a quest, encounter, or npc appearing in your party camp - will appear to direct you to it.