What weights are you referring to ?
Obviously newborns need to develop and take in stimulus before they "know" anything, so the initial conditions are not sufficient, but they are obviously necessary to make the limited learning useful.
As far my understanding goes to very large degree it is unknown how to model such dynamics, where it would be possible to start with no spiking neurons and evolve effective/stable learning behavior from small number of examples/experiences.
There may be many roads to intelligence. The path biology and evolution took may just be one such path.
Are they coded in DNA, or do epigenetic factors, the environment cells grow in, adjacent cells, their interaction, etc, play a role here as well?
Does this refer to DNA -> MRNA -> to protein or is there some other mechanism here?
Also, since you asked, I would like to mention protein interactions with cellular components and tissues. These can be argued as just proteins interacting with each other but the extreme complexity of these higher levels and the unique phenotypes arising from them make me think they are deserved to be treated as another layer of "data decompression".