I need to pack a texture into a bit array. I now have a bunch of pixels, each is either ff000000 or ffffffff. Looks like a good place to compress.

What is the better way to extract bits from a byte? Shifting left, and then doing "and 1"? Or shifting right, and then using the carry bit? What are the instructions for these?

Can I use SIMD here somehow?

SHR, SHl, RCR, RCL, ROR, ROL Yes, but there is no carry flag in the SIMD instructions.

Which SIMD instructions/intrisics?

The x86 CPU doesn't have a carry flag for SIMD or AVX instructions. Intrisics is just C's way of encoding the SIMD instructions.

if the least significant bit is '1' then shr will take off that bit and set carry flag to '1' so its not recommended to use that.
instead you can use rol and ror.

if you wanna use shr and shl without losing the least significant bit, then you need CC (condition code) like cmovpe cmovpo jpe jpo setpe setpo
pe = if parity flag is even
po = if parity flag is odd

even if you lost the LSB and carry set to '1' you can move it into a register.
there are many choices tho, depends on your need or how you wanna do your things.