I am wondering whether voltage is stable at the very beginning when fans and disks spin up (as their coils are almost like a short) and capacitors are filling up. How do ICs Deal with initial voltage drops? Do they wait for things to stabilize?
Yes. The designers of ATX specification have considered this problem.
https://en.wikipedia.org/wiki/Power_good_signal
On the motherboard power connector, there's a pin called "power good". The controller chip in a properly-designed PSU will determine whether the voltage rail has stabilized and only generates this signal when it did, the motherboard will only attempt to start when the "power good" signal is activated by the power supply, at this point all the inductors are saturated and capacitors are charged, so it's good to go for most boxes. Spinning fans are typically a non-issue. Also, the power circuitry on a modern motherboard has sophisticated control and protection, some can turn itself off or force-reset when significant over/undervoltage is detected.
However, if you have too many HDDs connected it could still be a big problem. When BIOS spins up the disk simultaneously, the inrush current of the motors is HUGE. In a good server setup, the disks have its own dedicated power supply, and a good disk controller will take care of it by not spinning all disks up at the same time. Some BIOS can also do that.