IOS 10 beta and profile

No. As explained in the documentation provided to you when enrolling in the BETA program you can update to the final release once it is available.

https://beta.apple.com/sp/betaprogram/faq

How do I get a shipping version of iOS back on my iPhone, iPad, or iPod touch?

Always back up your iOS device before installing beta software. To get a shipping release of iOS on your iPhone, iPad, or iPod touch, you can simply install the final version of the software you are testing when it appears in Software Update. If you would like to go back to a previously released version, you must first erase the beta software, then re-install the latest shipping version of iOS, and finally restore from your previously saved iTunes backup.
Learn more about how to erase and install iOS
Learn more about restoring from an iTunes backup

Software updates are usually released sometime between 10 am and 12pm cupertino time, which is in about half an hour.

Just be patient and it will be released.

I don't think he'll get the update today if he has the GM installed. It's the same build as today's official release.

Last year with iOS 9 there was no update from the GM to the official firmware as it was also the same thing.

exactly, if you were on the beta, the last update you got was last week and it's the GM. The gold master (10.0.1) is the exact equivalent of what's being released today... so you get no update today. Now if you want to leave the beta program (for future versions, 10.1 etc...), all you have to do is delete the beta profile (general/profiles) AND reboot.

http://appldnld.apple.com/iOS9.3.5/031-73998-20160825-34E22EF6-6715-11E6-A9F0-5D 3E34D2D062/iPhone8,1_9.3.5_13G36_Restore… iOS 9.3.5 restore

<This is a Direct Download Link>

After installing the latest ios 10.0.1 version, should I delete rhe Public Beta Profile?

Only if you don't wish to receive any further beta versions in the future.

If you may want to install other Beta versions when they are available in the future, you can keep the profile.