In what ways does a Commander in their Command zone influence the game?
Only Oloro's third ability functions in the Command Zone, and none of Gahiji or Sydri's.



Abilities of permanent cards usually only work on the battlefield, not in the Command Zone.


  112.6. Abilities of an instant or sorcery spell usually function only while that object is on the stack. Abilities of all other objects usually function only while that object is on the battlefield. The exceptions are as follows:


It is notable that characteristic-defining abilities (e.g. Ghostfire's first ability) function in all zones, and even outside of the game.[112.6a] The rest of the exceptions are quite intuitive. For example, if an ability can only be activated in a particular zone, it can be activated in that zone.[112.6i] And if an ability can only be triggered in a particular zone, it can trigger in that zone.[112.6j] This exception is what permits Oloro's third ability to function in the Command Zone.

But none of the other abilities of the three cards you mentioned meet one of the exceptions that would allow them to work in the Command Zone.