Id like to ask what are the requirements of registering an ICO?
It depends on your jurisdiction, but IMO you won't need ICO for your project.
Let's say my project won't produce any crypto currency, but needs some voluntarily developers instead of investors who whould be interested in using the created platform for themselves instead of getting any money profit.
It's basically an open-source project.
If it's possible will it be okay to make the number of tokens equivalent to the number of developers participating in the project's development and to give one token to one developer?
Yes, but that is not needed. The token will be worthless just like karma, merit, etc.
If the project can be useful then the devs will be benefit from it anyway, it adds good experience on their resume.
And will such an ICO protect the project's idea of being stoled and used by other people or there's no such a protection and it will be necessary to get some kind of patent?
MIT License? Anyway, before you continue, you should ask yourself if this project will focus on earning profits (commercial) or not.