Do you have any problems, i.e. things not working? If not, I don't think there are permission issues. The  open_basedir php variable not being used on your server  is actually good if you ask me as it just causes trouble if access to directories is limited by using this variable.

The reason the other listed tools are not working has nothing to do with permissions. First to note: these are not required for your site to work but if available they can improve performance (in some cases a lot!). "Unavailable" means that these tools are not installed on the server. And in addition to the installation they would also require some configuration being done.

My suggestion would be to ask the support of your webhoster if they can help you with the installation. For OPcache it's probably only necessary to install the opcache php module and then set a few variables right in the php.ini configuration file on your server. There's no need to configure anything in Elgg for OPcache to work (if available, Elgg uses it).

Memcache requires the memcached package (memcached and not memcache - the latter also exists but I believe is no longer maintained), libmemcached library and memcached php module to be installed. Additionally, memcached must be configured on the server and if memcached is running then Elgg's settings.php file needs some config variables being enabled and maybe adjusted. It's not too difficult but not trivial either to set memcached up, so if the support can help you, let them help you.

I never used Redis. As far as I know it's similar to memcached. I don't know if using both makes sense or if you can use only either Redis or memcached.