You could be experiencing one of two things. One may be some unkind latency. The other is a misconception about Hammer Slam. Hammer Slam's high damage zone is not the center of The Rook. It's the center of the hammer where it lands. The instinct to run away from The Rook during a Hammer Slam may very well keep you in the high damage zone longer. Strafing and running through tend to be better tactics to counter.
What I can assure you of is that Hammer Slam does not gather targets to damage until the cast is complete, which is around a second.
Rain of Ice is a near-instant cast, so I wouldn't be surprised if the damage goes through faster than expected. I can always double check and see if it makes sense.
Fireball is a different matter. Unlike Rain of Ice and Hammer Slam, Fireball is targeted on a unit. When the cast completes, a projectile is created, the target is set to the unit targeted, and away it goes. I suspect the tricks that break Snipe's targeting would break Fireball's targeting. Fireball is faster cast than Snipe and has less of a tell that it's happening.
Abilities and items with a cast time can be stopped. A well-timed stun, interrupt, or silence will stop the cast and put the ability into cooldown. Almost every Demigod gets access to an interrupt, stun, or silence in their skill tree (QoT doesn't have one, Regulus is left up to chance).