Apple - Why do iOS devices take such a long time to turn back on?

Like you said in a comment, it's more a software limitation. Even though there's probably some hardware safeguard.

If you plug your iPhone a millisecond before it reaches the shutdown limit, then, no harm done, it doesn't need to shut down, power is there to keep the phone running and recharge the battery.

If you plug you phone after the shutdown limit, then the limitation is in place to ensure things will go well the next time you power it up. If it allowed to power up the phone any time, then what if it is unplugged while booting. There wouldn't be enough juice for the phone to be fully back on and it would shutdown while booting.

That's something that could damage the phone or the OS.

So to ensure that this wouldn't happen, Apple probably put the arbitrary 5% limit.

This is because any type of rechargable battery needs a small amount of juice to recharge. If you drain the battery completely, it will not be able to charge again. This is true of lithium or regular rechargable batteries. The delay is there to ensure you don't damage the battery. If you were to completely drain regular rechargable batteries and put them in a device that runs on a tiny amount of juice until there is absolutely no charge, they would fail to recharge as well. In fact, I've had this happen. The reason why the delay is so long is because if you were to turn the device back on with a very small charge and ran software (say a 3d game) that uses a lot of power, your device may drain too fast to maintain positive charge and not be able to charge any longer.

Well, I can only answer part of your question, and that's the "Also if I plug the cable in less than a few second after the screen goes to the spinner, why does it take so long?" part: When it reaches the "spinner" screen, it's already begun the shutdown process. It's not like it's an OK-you-really-need-to-plug-me-in-now-this-is-your-last-chance screen. When it finishes shutting down, then it needs to charge to 5%. Then, you can turn it on.