Is it possible for a singer to sing a song just by reading its musical notation? A song which he/she has never heard before?
Yes, a properly trained musician can sing music (either in their head, what we call "audiation," or out loud) just by reading the notation. This is something that we call "sight singing," and it's a standard part of musical training.
There are a lot of problems with the metaphor that "music is a language," but the metaphor does apply here. Think about a language you can speak fluently: if you see it written down, can you reproduce it in sound? Furthermore, if you hear it, can you reproduce it in notation?
The same logic applies to musicians: sight singing translates written notation into sound, and what we call "dictation" translates sound into written notation.
With that said, musical dictation is rarely as accurate as language dictation, if only because there are often multiple lines happening at once in music as opposed to just a single thought being expressed. When you see (or hear of) musicians hearing a full orchestra and immediately being able to transcribe every musical line, these are savant-level dictation skills; very few musicians have that ability.
Yes. This is called "sight-singing", and it is a skill that can be learned. Musicians trained in sight-singing typically need to be given a starting pitch and can then sing the song accurately from there.
There are many posts on this site about learning to sight-sing and to train your ear as a part of that process. Search for questions with the tags sight-reading and/or ear-training. That will give you a start.
It doesn't even have to be a singer, many instrumental players can do this - particularly the tune itself.
It's a very useful skill for any musician to learn, especially for singers. Knowing intervals is key to being able to do it - and it does save an awful lot of time in rehearsals.
A lot of composers will be doing a similar trick while composing - not singing out loud, but hearing the different parts in their heads.
I used to work with vocalists who would turn up at gigs and simply read songs they'd not seen or sung before. Really it's hardly any different from instrumentalists sight-reading well. I guess it would help if a someone possessed absolute pitch, although having a good relative pitch would do the job, the vocalist having a start note or key chord.