The GD Library should be available under PHP 7. As for the MySQL version, we currently have the latest patched 5.5 release installed, but will likely upgrade to at least MySQL 5.6 in the future as various hardware and software upgrades are scheduled. (Wordpress itself will run fine on MySQL 5.5, they just recommend 5.6 for better performance in some cases.)
Concerning shell access, you should still be able to login to shell.sonic.net over SSH using your sonic username and password. This will allow you access to your web hosting files (under the public_html folder) if you prefer to work with them through a shell environment. You won't be able to add PHP libraries, but you can turn on/off PHP features through the use of a .user.ini file.
-- Joe M
Sonic System Operations