#1
Heyho

Does this scale has a special name:

C D E F G Ab B

or

1 2 3 4 5 b6 7

Sounds sweet descending.
#2
The fifth mode of lydian diminished (1 2 b3 #4 5 6 7) if you really wanted a special name.

I'd just call it ionian b6.
#4
Yeah that's a C Harmonic major scale
Bands I've seen live:

Def Leppard
Journey
Tool
Testament
Megadeth
Slayer
Mastodon
Deftones
Alice in Chains
Foo Fighters
Rodrigo y Gabriela
#6
Quote by Dayn
The fifth mode of lydian diminished (1 2 b3 #4 5 6 7) if you really wanted a special name.

I don't think the scale you wrote is Lydian diminished.

Lydian Diminished has BOTH a #4 and a b5.
Not possibly in eqal temperament.

To make this post relevant to the TS, Isaac has it correct.