how can I send emails from firstname.lastname@example.org (without any "behalf")?
Whether the "behalf" (or "sent via") is visible is dependent on the email client receiving the email, however, it can always be dug out by viewing the source of the email. You get the "behalf" when using Gmail's SMTP server (the default) to send the email. Gmail will add a
Sender: header with the gmail address of the account actually sending the email. This is correct behaviour as far as I'm aware and helps prevent your email being blocked as spam.
To remove the "behalf" when sending from Gmail, you need to use an alternative SMTP server - preferably one that is associated with the account/domain through which you are sending mail. This is what you will need to pay for. However, this should not cost very much... any hosting/email provider should give you this facility. A one off fee ($10-$40 dollars perhaps) for as many user mailboxes as you require, but limited by overall storage.
(Gmail for business charge $50 per user/mailbox - but provide a lot more than you seem to require.)