#1
Is C6 the same as Cmaj6?

I know Cmaj6 is 1, 3, 5, 6 but is there such thing as C6 built differently?
#2
Chord wise they are the same thing, but by using each particular name, the composer might be implying a certain scale the chord is taken from. Cmaj6 might be implying a major 7th betwene the root and 7th degree while C6 is probably implying a minor 7th (mixolydian). However, most of the time it's probably the same thing.
#4
I didn't say that you'd have a 7th in the chord. I said the composer may choose Cmaj6 or C6 depending on what mode he wants played over the chord which would most likely affect the 7th.