For example, can I rename system log and or delete some of the log buttons on the toolbar?

All KDE applications support modifying their toolbar by default, unless restricted by their developers ( as in System Settings )

Simply right click on the toolbar and select "Configure Toolbar"

The part that I am trying to modify seems unmodifiable, it's called
Actionlist:toolbar_log_mode_list and it says that if it's removed, it cannot be re added.

Those are plugged action lists, and correct, it is not possible to restore them or change their state, except for programatically. Please file a feature request report at bugs.kde.org if this application is part of KDE Trunk, or with the appropriate application method if hosted in Extragear or KDE-Apps.