In these sorts of cases, I trust the community of professional and semi-professional musicians more than my own music history knowledge. (Read: I searched youtube for several videos and saw what they did).
For the uninitiated who stumble upon this question: a mordent is the proper name for the squiggle above the staff in measure 4. It is similar to a trill, but starts on the note in question, and only alternates with the note above it once.
In every instance I watched (a different piano solo arrangement, an orchestral arrangement, A 4-hand piano arrangement, and a piano/violin arrangement), the initial mordent is played between the D and the C. Most of the videos play the mordent before the beat, with one playing on the beat. Most of those videos included both the top and bottom notes in the mordent. Further support for including both notes comes from observing the grace notes in measures 6, 7, 10, and 11; it seems like the intent is to have both notes involved in the ornaments.
If you can play a sixth with a 3-1 stretch, I would recommend playing the mordent with 5-2 and 4-1, playing the third diad with 3-1. Keeping the mordent light will help you play it quickly. If this is too difficult, it's acceptable (especially below a professional level) to involve only the top note in the mordent.
It's worth noting that many piano teachers advocate first learning the piece without any ornaments, and then adding them in at the end. This is largely to avoid any timing issues, as these ornaments typically should not alter the timing of the other notes. If you notice that any ornaments are bogging you down or screwing up the timing of the other notes, it might be worthwhile to leave them out for the time being.