The common convention is that women's parts are written in the treble staff, and men's parts are written in the bass clef. However, a somewhat common exception is that when you see a part written in the treble clef, which is to be sung by a male voice, it is taken an octave lower (the exception to this exception is countertenors, who sing treble clef at written pitch, in the same range as a woman). Sometimes this octave lowering is made explicit by adding a small 8 under the treble clef, but this is not always the case. This octave lowering works because, in general, men tend to sing about an octave lower than women.
So why notate in the treble clef instead of the bass clef? Simply because it makes the music more accessible. If a female vocalist, or any number of instrumentalists, wanted to play the melody line, notating it in the treble clef will be easier for them to read than the bass clef. Also, especially at the amateur level, it's more common to find people that can read the treble clef down an octave than can read the bass clef up an octave. Treble clef is something of a lowest common denominator among musicians.
In your case, the score is more or less accurate in terms of pitch and key (granted I haven't exactly proofread it note by note). As notated, it's range is G4 (G above middle C) to A5 (above treble staff). Using the convention above, though, where male vocalists take the treble staff down an octave, it's "concert pitch" range is G3 (G below middle C) to A4 (above middle C).
As you point out, the higher notes are rather high in the tenor range (though not uncomfortably so), yet they don't sound like what you would associate with a typical "high guy" voice. You ask about the vocal timbre, and you're spot-on with your question. I'm not a vocal expert, but I believe the singer is using a technique common in pop music, known as "belting". While voices typically have a lower "chest" voice and a higher "head" voice, my understanding is that belting is a forceful technique that produces a chest-like sound in the upper head-voice range. There's a lot written on both sides about whether belting is a dangerous technique, and how to do it properly. I'm not going to go into it any further because I don't know the details, and it's out of scope for this question (but it would make a great new question here).
One final thought: if you listen to the coda (~4:20), you can hear the difference between the belting timbre that was being used earlier, and a more regular sounding "head voice" sung in the same range. Because belting tends to produce a louder sound, and the coda is quieter, he switches to a head voice and you can hear how high the notes really are. If this were written an octave lower, the coda would not have nearly the same strained quality as it does.