LoginToBoggan + Don't use set password mail I would like to give my user the opportunity to register and login with username AND email. 

Therefore I installed the module LoginToBoggan where you can enable that users can login using their e-mail address.

When you want users to set their password at registration you need to check the checkbox Set password. But when you do this you also get a mail to set your password (can only be used once).

But I don't want that. I just want them to register with email/username and directly login in without getting ANY email. I fire a rule to send a mail after I created an invoice so I don't want the drupal account emails.

But I didn't found out how I can simply do this?
I use the same type of setup as the one you describe:


Register and login with username and email.
Login using the email address.
User sets password upon registration (i.e. no email confirmation required).


I've found that the process where user receives a one-time link to set password is a huge stumbling block in the registration process.

Drupal let set up the site to have user sets password upon registration (Uncheck "Require e-mail verification when a visitor creates an account" on Administration » Configuration » People » Account settings).

However, I do not use LoginToBoggan to allow the users to login using the email address.

Instead, I use Realname Reg.  Please note that this is a sandbox project, but it is actively maintaned (by me), and it has been tuned specifically for the use case you describe.  I've used it on a dozen production sites for about a year with no issues.