This is the beginning of Oxygene part IV (by Jean-Michel Jarre). There is a note-equivalence notation (not sure what it is actually called). I do not really understand how this should be applied to figures other than the dotted quaver + semiquaver. Especially, I am not really able to figure out how the triplets should be played and how they should be played in relationship with the left hand dotted quiver + semiquaver.
Answering your first question: the triplets should be played exactly as triplets.
About the rest, Tim's answer and the comments tell you everything else.
One detail: The G after the C under "Trb" (last bar, right hand) is notated as a 16th note, but should also be played as the last note of a triplet. In other words, C-dot-dot G may be thought of as a quarter note C, tied to C-dot G, where the latter is another instance of a dotted tuplet to be played with a swinging triplet feel.