How to edit the apple developer account

As mentioned on iTunes Connect:

To setup a new bank account on iTunes Connect, take the following steps:

1. Access the Contracts, Tax and Banking module
2. Click on either the "Set Up" or “View/Edit” link in the Bank Info column
3. Click on the "Add Bank Account" link
4. Select your bank country and click "Next"
5. Enter the relevant bank code (local clearing or Swift code) or look up your bank, and click "Next"
6. Confirm your bank selection and click "Next"
7. Enter your bank account information and click "Next"
8. Review the confirmation screen, select the check box at the bottom of the screen to certify your information, and click "Save"
9. On returning to the bank account selection screen, select your new bank account from the drop down list, and click "Save"

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?).