Are there any other special requirements for three sounds to be a chord by musical definition? Do overtones or the same note just sounding on different octaves (sounding together) still can be named a chord?
We wouldn't refer to this as a chord. The idea is that you need three unique notes, which would not include octaves. Some definitions of a chord will allow just two notes but the more commonly accepted definition calls for three.
You could, however, imply a chord with octaves alone, which requires context. For example, if you have established a key by playing a few chords or enough melodic notes, playing a single note, or octaves, within that key will be able to imply which chord and of what type the single note would be harmonized with. A lot of rock music uses what we refer to as "Power Chords", which consist of a root, fifth and octave. A single power chord wouldn't give us enough context to determine a key because a fifth is the most common interval you will find within a given key, as all but the 7th degree of the major scale have a naturally occurring fifth. However, when a sequence of power chords is played, you can usually determine the key/tonal center and be able to infer the chord type (major or minor) that each power chord would represent.
A chord is defined as a set of pitches. If you are really into semantics you could argue that the notes C4, C5 and C6 meet that requirement and could be considered a chord, but to most people it would not be a chord.
There are two big reasons for this. The first is whenever someone looks to define a chord, they look at the collection of pitches by the actual note not by the octave it is in. It's why we view the set of notes C4, E4 and G4 the same chord as the set of notes C3, G3, E4, C5, and E5 and give it the same name.
The second reason is we use chords to describe harmony which just one note octaves apart most would say does not count as harmony and would consider any piece that has lines in unison or an octave apart being monophonic (one melody, no harmony). Going along with this, when we do build chords we typically like to describe them as 3 or more distinct notes that imply some kind of harmony although two notes can imply harmony like C and E we could imply as a C major chord.
So keep both points in mind when trying to describe a chord.
2Just to expand on your first sentence: a chord is often defined as a set of pitch-classes, rather than pitches. The difference is exactly that pitches differing by octaves are considered the same pitch-class: i.e. C4, C5, C6 are three different pitches, but are all the same pitch-class, C.– PLLMay 10, 2016 at 5:22
This would better described as tone tripling than as chord since the question is dealing with notes separated by an interval of one octave and each note is basically twice as fast as the previous in frequency. The Harvard Brief Dictionary of Music states "... the octave is the most perfect consonance, so perfect indeed that it give the impression of being a mere duplication of the original tone." For this question we would say tone tripling. These means we get more power from the same but very little harmonic interplay which is really the basis of all basic Chord Theory.