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 ...
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 ...
Gentoo is a source-based distribution, where everything is built from the source. You can tune GCC to build for i586 like this:
COMMON_FLAGS="-march=i586 -mtune=pentium -O2 -pipe"
in your /etc/portage/make.conf
Changing -O2 to -Os will decrease executables size a bit.
Another thing you need to worry is to build correct kernel.
If the build process ...
The original Pentium, which succeeded the i80486, was indeed the original definition of the i586 instruction set - and not x86_64. But the Pentium was introduced in 1993, and your HP machine is much newer than that. Intel has continued to use the Pentium brand for almost every x86-compatible CPU line they've made since then, to great confusion among people ...
I don't think your scheme is any more extensible than Intel's. It's easy to see how to widen the address space in Intel's scheme: just increase the shift value in later generations.
Of course, this will break software that tries to do arithmetic on segments assuming that the addressable space is and always will be 1 MiB, instead of treating them as opaque ...