EASY SOLUTION -
To properly set Gmail to handle your domain email accounts in from within your personal free Gmail account, follow these exact steps in order:
1. In your cPanel > Email Accounts create the account firstname.lastname@example.org (replace email@example.com with your actual email address) and make note of the password you assign it. NOTES ON THIS STEP - Set the quota to 80mb or lower, since you won't be leaving emails on the server and Gmail will synchronize automatically between all your devices via Gmail's Webmail interface on your computers and the Gmail App on your phones and tablets. The quota will serve as a handy warning if something stops working correctly down the road as well. Never set the quota to unlimited or you could accidentally fill your hosting account with emails and that would cause other issues.
2. Log in to your Gmail account and go to Settings > Accounts and Import and in the "Check mail from other accounts" section click "Add a mail account" and provide it with firstname.lastname@example.org (replace email@example.com with your actual domain email address) then click the "Next" button
3. In this next dialogue box, put firstname.lastname@example.org in the Username box and provide the password that you assigned in step 1 above, and click "Next".
4. Select the "Import emails from my other account (POP3)" option and click "Next".
Now you will be asked to enter your domain email account info as follows:
Email address: email@example.com
POP Server: mail.example.com (replace example.com with your domain)
DO NOT PUT A CHECK IN THE "Leave a copy of retrieved message on the server" option. Leave it UN-checked. In fact, you can leave all the check boxes blank.
Click the "Add Account" button and you'll be asked if you want to be able to send email from that address.
5. Make sure "Yes, I want to be able to send mail as firstname.lastname@example.org" is selected, and click the "Next" button.
6. Enter your desired name in the Name field (for example - John Smith - this is the name your recipients see on emails you send) and be sure to UNCHECK "Treat as an alias" and then click "Next Step"
7. In the SMTP Server box - change it to mail.example.com (replace example.com with your domain) and leave the port as 587. Now in the username box enter email@example.com and provide the password that you assigned in step 1 above, leave TLS selected, then click Add Account
8. IMPORTANT - Within 5 minutes (if you did everything above correctly) a Gmail Confirmation to Send Mail As firstname.lastname@example.org message will arrive in your Gmail account with a link to click and Confirm / Verify your domain email address. The email will also contain a verification / confirmation code, but if you simply click the confirmation link and hit "Confirm" button, then you don't need to enter the code anywhere and you can simply close the "Confirm verification" dialogue box. Either way - make sure you've verified the address with Gmail before you go to the next step.
You're now using your free Gmail account as a mail client for your domain email account, and you can send/receive your domain emails right from within.
Note 1 - The above steps will also automatically add your domain email address to the "Send Mail As" list in your Gmail Accounts and Import settings.
Note 2 - You will probably want to make sure that the Gmail > Settings > Accounts and Import > When replying to a message setting is set to "Reply from the same address the message was sent to"
Note 3 - IMPORTANT - Depending on your incoming mail volume, Gmail might only check for new messages on your domain account every 30 minutes. This is not something you or we can control - Gmail automatically adjusts the checking frequency itself based on your incoming volume. You CAN manually check for new messages from within your Gmail account by going to Settings > Accounts and Import in Gmail and in the "Check mail from other accounts" section click "Check mail now" to manually check for messages on your domain address.
Note 4 - Mobile Devices - On your phones and tablets, install the Official free Gmail App using your personal free Gmail account settings, and you'll see that you're able to handle your domain emails from there when you're not at a computer!
If you follow the steps above correctly - you will now be able to Receive AND Send from your domain address from within Gmail and will automatically remove messages from the server so the hosting account Inbox will not fill up, and everything will automatically be synched with any device that you use Gmail web interface (computers) and the Gmail App on (phones, tablets).
Looks like a lot, but the steps are actually quite quick and easy. Once you've done once or twice you likely won't even need these instructions!