I have access to some writer's guides and various notes. There's nothing in the ones that I have regarding the exact process by which the Augments were created.
All we know is what is already told to us in the episodes themselves: the Augments from Cold Station 12 were embryos left over from the time leading up to the Eugenics Wars in the 20th Century.
There, is however, some useful information in Memory Beta land (that is, in the novels and comic books):
The origins of the Augments were said to had been traced to a group of international scientists, such as Dr. Heisen, who worked during the 1960s-1970s who had laboratories in Haiti and Pakistan as well as Chad along with their headquarters in North Yemen. These progenitors of a super race worked in the poorest countries on Earth as it was easy to bribe the officials into silence, take and hire human guinea pigs for their work. Unknown to him at the time, but Ralph Offenhouse supplied this cabal with equipment from electron microscopes to computers, drugs and even chemicals which broke export regulations and smuggling laws. (TNG novel Debtors' Planet, TOS — Khan comic Issue 1)
This led to the culmination of augmentation amongst the Human race through the production of the Chrysalis Project in the 20th century on the planet Earth...
Some people and/or Augments did indeed escape Earth during and after the Wars, often ending up supporting Eugenics anyway. (TOS comic Sceptre of the Sun)
The Earth authorities were later responsible for confiscating the remaining Augment embryos and placed them into storage. By the 2130s, they were stored in an Earth Starfleet facility known as Cold Station 12. (ENT episode "Cold Station 12")
Let me try to comment on your questions now, in order:
- With regards to the talk of "brothers", was there a blood relationship between Augments?
According to the summary above, the Augment embryos were created using DNA from human guinea pigs in poor countries. Whether or not the donors were related to one another is unclear, but there is nothing to suggest so. The term "brother" seems to function purely for the promotion of camaraderie amongst Augments.
- Did people simply volunteer to donate egg and sperm?
As mentioned above, the scientists would "take and hire" the donors, implying that some were compensated (likely a token amount) and others were taken against their will. In such impoverished places, guinea pigs could disappear with little suspicion.
- Was there already a fetus and it was removed from a mother's womb?
The term "embryo" and the term "fetus" are used at different stages of development. A blastocyst is called an embryo until the 8th week after fertilization. After the 8th week, it is called a fetus. In other words, the terms are mutually exclusive. Since the Augments are specifically indicated as arising from embryos, there were no removals of fetuses from mothers' wombs. It would seem that the embryos were created / modified in vitro.
- How did they figure out which humans to start with?
This is unclear, but it seems that the leaders of the Chrysalis Project were more concerned with concealing their efforts than with the quality of donor material. Donors, willing or unwilling, were picked up off of the streets in the aforementioned countries, and the only concern was bribing government officials so that they wouldn't "notice" such activities and abductions. In other words, the project leaders were likely more concerned with the finished product than with screening the source material. Perhaps the scientists themselves would have appreciated having source material that was screened and which was of the highest possible quality, but they didn't seem to have that luxury.