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If I move to consecutive frets on the same string, are notes produced from the same octave?
For example, is it possible to play the first 5 notes of C major scale on the A string? (C, D, E, F, G)? Why do we play D note on the open string instead of moving down the same string?
Yes, you can play C major scale the way you show, and there is nothing wrong with it. Actually there is some benefit, as notes on different strings sound slightly differently (e.g. compare sound of D on the open D string, and D on 5th fret of the A string), so playing on a single string gives you the most consistent sound.
But there are also downsides: you need to move your fretting hand a lot, and you are limited in how high or how low the melody can move. It complicates even more if you want to play several notes at the same time, like chords, or polyphonic lines. These are reasons to split melodies onto multiple strings.
I'll also answer the title question, even if it's perhaps not worded the best: The convention is to name C as the lowest note in each octave. So e.g. C on the 3rd fret of the A string is called C₃; then D on the 5th fret is D₃; up to B on the 14th fret is B₃, and then C on 15th fret is C₄, it starts next octave. Similarly B on the 2nd fret of A string is B₂, so it belongs to the octave below C₃. That's naming convention. If you play C₃ and you want to move half tone down, you play B₂, and there is nothing wrong in the fact that you move to an octave with a lower number.
I’m going to add some useful information about the guitar to the capable answers already provided. One of the characteristics of the guitar is that with the exception of the lowest 5 notes on the low E string and the highest 5 notes on the high E string every note on the guitar exists in at least 2 places on the instrument. Most middle register notes are available in 3 to 5 different places. As a matter of fact, if you have a 24 fret guitar the open high E note actually can be played in 6 different places! Of course some are much better sounding than others, for example the 24th fret of the low E isn’t going to sound very good.
What this gives you is the option to play a scale or anything else in many different ways by using one string or switching from one string to another. Some are more logical than others and this is something you learn with education and practice.
Let’s take your example of C D E F G. Here are 4 examples of ways those notes can be played on the guitar (there are several other ways to play this).
There is no right or wrong way and there are other ways as well. Usually the context or what comes before and after will determine which is best to use.
ADDITIONAL INFO:
A guitar is tuned in 4ths, which is the equivalent of 5 frets. The exception is the G and B strings are tuned a major 3rd, or 4 frets apart. The 5th fret of the low E string is A, the same as the open A string. The A string 5th fret is D, same as the open D string, etc.
If you study the fingerboard diagram you used in your question you will notice something. Any note has its equal note either 5 frets higher on the adjacent lower string OR 5 frets lower on the adjacent higher string. (Except for the G and B which would be 4 frets). If this isn’t clear I will give an example:
Take the note E on the A string, 7th fret. That note is duplicated on the E string 12th fret (one string lower and 5 frets higher) and also on the D string 2nd fret (one string higher, 5 frets lower).
You can look for other examples on your own. This will help you learn the fingerboard little by little. It may seem overly complicated but in the end it actually makes sense because the sequence of notes is always the same, it just repeats in different positions on different strings.
Short answer: yes. You're right. You can play almost 2 octaves of a scale on a single string (maybe more if you're bendy and have an electric guitar).
The reason not to play on the same string is that eventually, you'll run out of consecutive fingers to use and you'll have to move your hand laterally. That will make it harder to play the notes smoothly, especially if playing fast.
Musically, the change in string does affect the sound tone as well, but that's not really the main reason for string choice.
Why do we play D note on open string instead of moving down the same string ?
Sometimes we do, sometimes we don't.
There are many ways to play a C scale, and the choices go to preferred style. For example, a folk style would like ringing open strings in the first position, but jazz and rock players would play it higher up the neck in part because it eases muting.
Playing that sequence up the neck would indicate, to me, a position change that would correspond to a change in style and tone, like many rock solos going higher in pitch to increase intensity.
