Yup, the permissions notice is a bug. You can ignore it until I publish Akeeba Backup 3.6.3.
Regarding PHP, I put that notice in place because I plan on dropping PHP 5.2 support sometime around Christmas. OK, the notice is way too prominent. I'll make it less visually disturbing in version 3.6.3, but it won't go away. I want everyone to have a fair warning that I will have to drop PHP 5.3 support. At some point even my development machine will be incapable of running PHP 5.2. Since I won't be able to ensure I can properly test with PHP 5.2 I will have to drop support for it. I'd hate to give y'all a nasty surprise without fair warning.
Is the deadline stressful? Yes. Should it be? NO! PHP 5.2 is end of life since January 2011. I believe that this way too long for a hosting provider not to have figured out a way to support the currently supported PHP version (5.3) across all of their servers. Can you imagine me insisting that you should still use Joomla! 1.0? That's pretty much what your hosting providers are doing with PHP 5.2. If your hosts ignore you, vote with your wallets, people. It's not a matter of some developer waking up one day and deciding that PHP 5.3 is the only way. It's a matter of security. PHP 5.2 is ancient, unsupported and already has one very high profile, easily exploitable vulnerability. It's irresponsible and insecure continuing using it. It's as simple as that :)
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!