I'm getting Trials but not the Origins or Doom packs.

Will I be restricted in what created tracks I can play?

Let me clarify: without buying Origins of Pain or Riders of Doom, are there custom tracks I can't play? Like if a track is made using Riders of Doom objects, would I be able to play the track? If so, how do you know what tracks you can or can't play?

Or is it like LittleBigPlanet, where all user content is able to be played even if you don't have the DLC content that was used to make the level?

You cant play, there are little icons indicated if they are DLC tracks in the user created tracks, and you can sort them by DLC or Non DLC

Yep. You need to buy the DLC to play created tracks with DLC in them.