4a. False. Flexible diamond doesn't make any sense. Diamond is sp3 bonded carbon and those bonds are highly directional. They're not going to flex.. Metals are flexible because metallic bonds, unlike covalent bonds, don't confine the electrons in space. Whatever this purported carbon fiber is it either won't be flexible, or it won't be diamond.

Well you can bend diamond, you just need to be fast about it, like 310 kHz fast. But yeah, "thread" it is not.

Proteins can make and break diamond bonds

Well, this sounds a bit less outlandish if rephrased: "Proteins can make and break carbon-carbon sigma-bonds." Which of course happens all the time in the course of making organic molecules.

For making diamond specifically, you might try starting out with formaldehyde, and releasing water as you stick carbons together. Looks like there should be plenty of spare energy to do it. Hm, but that seems very tricky to do, I wonder how people do it with synthetic diamond. Ah. Very high temperature carbon radicals sticking to a lattice-matched substrate.