Intel 8008 CPU has an internal stack, implemented as an 8 x 16-bit scratchpad.
No, it's entries are 'only' 14 bit long, as all addressing on the 8008 supports only 14 bit (*1).
How does it work exactly? Is there any "invisible 3-bit stack pointer"?
Yes. In reality it's not a stack pointer, but selects the active PC.
A three-bit address pointer ...
Datasheet for the Intel 8008 CPU mentions that the Carry (C) flag is affected with the logic operations (AND, OR, XOR), but it does not make any sense. I believe Carry will be zeroed, but I have no proof of this conclusion.
First, the manual does not say anything special about logic operations but states:
The result of the ALU instructions affect all of the ...