Share iPhone developer account for multiple devices

Yes. Although I couldn't find any specifics regarding the possibility of a Company having two or more iOS Developer Program accounts, in the Enterprise Program FAQ it is stated that:

If I am enrolled in the iOS Developer Enterprise Program, can I distribute my apps on the App Store?

No. The ability to distribute apps on the App Store is only available to developers enrolled in the standard iOS Developer Program. The iOS Developer Enterprise Program only allows you to distribute your apps to employees or members of your organization through Ad Hoc distribution.

If your company is enrolled in the iOS Developer Enterprise Program but would also like to distribute your app(s) to customers on the App Store, you will need to enroll in the iOS Developer Program.

Which means that Apple has no problem with a Company having more than an account, which is confirmed (with a regular program too) by my personal experience.

Even more, in the specific case you present, as each developer would need to use his personal legal name, it would be, on practical purposes, as if you had hired a developer who was already enrolled into the program.

Is it better to create new Apple ID for my developer account, or is it ok to use my existing one?

I would recommend creating a new one. Apple doesn't seem to care either way, but it's usually a good idea to separate the two accounts.

And what are the rules regarding names (like can I use nickname or I have to use my real one)?

You have to use your real name. This is for identification and since the agreement is legally binding, I wouldn't risk it (plus, do you really want DenverCoder9 to show up on your banking records?).