The [E] part is in C Major key
and I can't understand the C#m7-5 chord at bar 3
what is this chord ?
Where did it come from? and what is its function ?
I would call this (and the Db7 that follows) passing chords to get from I (of V, since we've modulated to C major) to V7 (of I, which is F major).
Peculiarly, this means we're just moving from C to C7, and the passing chords are obtained by just a few half-step tweaks, but they're untethered harmonically, helping to ensure that there's no stable harmonic sense until we get back to F. In other words: the fact that you can't find a clear harmonic function for this chord is a feature!
As far as I can tell, the C♯m7♭5 is basically just a variation on the Cmaj7 chord before it, but changing the note C to the note C♯, thereby changing the chord's quality and creating a really cool sound. I've heard this called a "slide progression" before, and I think this is the same idea. In terms of function, this adds some tension that can be resolved, or more likely, linked to a subdominant chord. Also, I'd have to disagree with your analysis of the B♭/C chord, I'd call that some form of a C7 chord with extensions (it functions as a dominant chord).