Setting everything as writeable (777/666) is not the safest option. Also using apache under mod_php in a shared hosting environment is not the safest option, you better had php configured with suexec or fpm or other similar technology. Elxis needs only repository folder to be writeable, everything else can be not-writeable and Elxis will function fine. The files you show me are not from Elxis. There might be from the additional extensions you have installed but not from Elxis core. You can search your server log files to find out how they were generated and you can also open them to see what they contain. Such issues are mostly a matter of you and your hosting company. I cannot provide you more help because I dont know what has happened to your site. You can read our security guide for Elxis (not for your web server) if you like.
datahellWhich were the names of the files you saw in elxis root folder and what was their contents?