Before anything else, you should take a good look at that fretboard map that Jim linked. I think you are confused about the tuning of the strings and their order. Assuming you have tuned your uke in standard tuning, the lowest tuned string is the 3rd one (counting from the bottom if you are holding the uke in playing position and are right handed), and it´s a C. The second lowest string is the 2nd string, and it´s an E four semitones above the 3rd string. Then comes the 4th, with a G 3 semitones above the 2nd, and finally the 1st, with an A 2 semitones above the 4th.
1) Does C has a ascending tone from "open" string C (from left) -> 3rd fret (1st string) -> etc.. to 12 fret (2nd string)"?
but 12 fret (2nd string) isnt it equal to open C?
Not exactly, the 12th fret on the 2nd string is an E, not a C. Let´s try to picture this as a wave or an arrow going from the headstock to the bridge.
The lowest C on your uke is the one on the open 3rd string. We´ll call that C4 (because it´s on the 4th octave of a piano).
Going one octave up from C4 we have C5, and it can be played on the 3rd fret of the 1st string, 8th fret of the 2nd string, 12th fret of the 3rd string, and 5th fret of the 4th string (see how it kind of looks like an arrowhead?).
Then the higher C is C6, and the arrow has to move 12 more frets towards the bridge. You don´t have so many frets on the 2nd and 3rd strings, so you can´t play it on them, but it is on the 15th fret of the 1st string and the 17th fret of the 4th.
Try to fret the 8th fret on the 2nd string and the 15th of the 1st string and play those two strings as well as the open 3rd in succession, when you hear it you will understand why they are not the same, even if they are all Cs.
2) How to play notes that are lower than middle C (3rd fret (1st string)) when we see them in the music sheet? E.g. Does lower A and B is played with 4th string (fret 2 and fret 4)?
This question is why I think you are confused. 3rd fret of the 1st string is not the lowest not on the uke, so you have a whole octave below that note that you can play. The lowest is the C on the 3rd string when you play it open. To be precise, that is the one called middle C (even if it´s the lowest on the uke, it´s in the middle of a piano keyboard, that´s why it´s called that).
If you mean how to play notes lower than THAT C, then the simple answer is you don´t. As Jim said, you can try playing them one the 4th string, but they will be higher and you have to decide if it sound good or not (it will depend on the song).
There is another option, which is to transpose the whole song. That means changing all the notes so they are higher and there is none lower than middle C. Most music notation software can do that for you.
3) from one c (open C) to the end of the C (fret 12), is there five octaves?
See my answer to your first question. There´s only two octaves on a ukulele (from C4 to C6; if your uke has more than 15th frets then you have two octaves and a bit).
4) How to visualize the fretboard all the sequences of the notes with respect to the piano keyboard? Is it true that the 4th string (from left to right) is the first octave, then 3rd string (from left to right) is the second octave and so on to the 1st string?
Visualizing sequences of notes on stringed instruments is the biggest challenge for those of us that learned our scales on a piano or other linear instrument. I´m still struggling with it. The thing is that most notes can be played in different places of the fretboard. Which one is best will depend on the song and just what is most comfortable for our hands. While you could do a C major scale on just the 3rd string, it´s probably a better idea to switch to the 2nd and 1st strings at some point. On top of the fretboard map Jim gave you, this forum post can help you too:
http://www.ukuleleunderground.com/forum/showthread.php?9272-Uke-Scales-Three-Notes-Per-String-Method
