Are all the notes in each box part of the same chord?
Knowing the make up of chords will help answer. Take the third box (bar or measure). G add 2. That's G B D and A. Those notes are there, but also there's an E and an F#. So, obviously, the answer is no. The last bar chord is Em7b5, made up from E G Bb and D. There's also F# C and A. Answer no again.
However, thinking past the question's wording, should it mean generally, the same applies. In most given bars, there will be notes from the accompanying chord, with often two or three passing notes, used as 'stepping stones' to get from one chord tone to another. Merely using chord tones will work, but the melody won't be as interesting, usually, if that's the case.
No. There's a C# in the first box, and that's not in the D chord (D-F#-A) the first box is labelled with.
There are also a D and an E in the second box, neither of which are in the F#m chord (F#-A-C#) the second box is labelled with.
Actually, you can assume that the topmost melody is harmonically ahead of the boxes by an 8th note.