These sections give you, the performer, some liberty to make some decisions and "randomize" some notes, but within certain boundaries. (Note, you might use the word "improvise" in discussing this, but usually it means something much more free, like completely making up a melodic "solo" over a set of chords.)
The first excerpt has the text note: "R.H. and L.H. [right hand and left hand] play at random any notes given." That would seem to mean: of the noteheads printed in that measure, select any of those notes at random. The preceding measure is similar in range and offers a model: the rhythm is eighth notes (more on that later), the melodic contour is uneven (that is, it "jumps around" between notes rather than arpeggiating straight up or down), and most notes are played singly, except for a couple of double stops. The note confirms this idea, "composer generally plays only one note at a time." This seems confusing—composer?—but it amounts to a suggestion, and the composer Mike Ring is referring to himself in the third person. "You do whatever you want using these notes—buuuut, I don't usually play them as block chords; I usually play single notes."
As for rhythm of this section: The notes used here start two measures earlier, with a mixture of quarter and eighth notes. The performance video makes it clear that these are not intended as a strict rhythm, but as a kind of notated rubato, played very freely. The accel. in the next measure means "speed up gradually," to the point that the next excerpt requires the text note "steady tempo; 16th notes," meaning "the gradual increase in speed has reached the point of 16th notes; stop getting any faster." The entire piece, by the way, seems extremely free in tempo, in a spirit indebted to the French Impressionists (and sports a Monet painting, and a movement titled "Water Lilies on the Seine").
"High D played at random anytime in measure w/ l.h." is pretty self-explanatory. The notehead with no stem is a high d. Play the other notes in the measure as normal with your right hand. At some random point, as you please, use your left hand to play a high D.
For the cluster of low pitches, marked "Use palm": I could be wrong, but it sounds possible to me that the performer has reached over the music desk of a grand piano and touched the strings directly. Presumably this would mean to strike or brush a cluster of neighboring low-range strings with your palm, and would take some experimentation with technique. If I'm wrong, then it's playing the keys as usual, but mashing a cluster of neighboring keys with the palm of your hand rather than individual fingers.
By the way, there is a lot of French in the expressive markings of this piece. I would recommend looking up anything you're not familiar with, like chantant (singing) or avec energie (with energy).
Finally: The composer himself has posted the piece, provides his email address, and welcomes contact. If you are interested in performing the piece or need continuing clarification, probably the best thing is to contact him directly.
