I know that there are some special interactions when Syndra uses her E ability with neutral monster like blue buff or red buff and throws it on enemies.
Is there a similar interaction when she grabs Daisy, the ult of Ivern?
No. The special interactions are unique to Ancient Golem and Lizard Elder.
Force of Will has additional effects when targeting the Ancient Golem and Lizard Elder. Throwing the former will refund 10 mana to Syndra upon cast and will reduce the cooldown of Force of Will by 1 second, while throwing the latter will apply the Blessing of the Lizard Elder burn to any affected units in the area of effect.
Regarding pets:
Enemy pets that passively have an effect on their surroundings will continue to affect the area around them while they are being held. For example, Annie's Tibbers will continue to inflict burn damage on surrounding units while in stasis.
More information on the Stasis effect:
A unit that is in stasis is unable to control movement, declare attacks, cast abilities, activate items or use any summoner spell for the duration, while also being invulnerable and untargetable for the duration.