Sessions folder ... uhhh, did you delete the folder or did you delete the contents of the folder? (said to remove the contents of the folder ... all the files contained therein ... not the folder). It should be re-created, but it's always safe to play it safe. Does the sessions folder exist?
Are you sure you are running 2.3.x? Those config items related to are found in mdl_config table of a 1.9.x.
First item:
$CFG->eaccelerator is set to true but the required functions are not available. You need to have either eaccelerator or turckmmcache extensions installed, compiled with the shmem keys option enabled.
The first line says there is an issue with eaccelerator which leads to the others ... config.php (which must be read every access) and login.php. Think that is the issue. Notice I said think ... don't run eaccelerator but apc in php.
Here's an old thread (talking about older versions of Moodle, but one will see almost exactly the same error message):
https://moodle.org/mod/forum/discuss.php?d=71293
Not sure one needs eaccelerator any more - *if running 2.3->*. Beginning version 2.3 Moodle has caching of it's own (MUC). Besides that, have read where eaccelerator is a problem with Moodle.
Is this server remotely hosted and you have no options/choices for using/not using eaccelerator?
Can you access, directly, the phpinfo page?
http://yoursite//admin/phpinfo.php
That might prompt you to login as an admin user but hopefully will redirect to show information about your php config.
The only other way is to have ssh terminal session into the server and as root user issue:
php -i
php -m
'spirit of sharing', Ken