Is there a possibility to log in with the user name
Of course there are, you can log into the Minecraft launcher with email / username + password.
Nope, only with email.
Provided the account is newer than 2012 or has already been migrated. Then it's a Mojang account and you need the email to log in.
No, it does not exist.
Depending on what you mean by "forgot your email", you have two options:
You have forgotten which of your email addresses you used: Click on "Forgot your password" and see in which of your emails the password change email arrives.
You have completely forgotten the email: Contact Mojang customer support and have your Transaction ID ready. https://help.minecraft.net/hc/en-us/requests/new
