Abilities with casting times are not interrupted by Rocket Grab (or any crowd control) and will go off as if Blitzcrank hadn't grabbed them.
For skillshots, such as Final Spark, they will fire from the point of cast regardless of the caster's position when the casting time ends.
For dashes and blinks such as Arcane Shift the caster will blink to the target location after the casting time regardless of crowd control effects. Blitzcrank's arm will visibly stretch and then fizzle.
Contrary to popular belief, you cannot cast Arcane Shiftduring Rocket Grab. However, as of V126.96.36.199, spells cast while disabled will immediately be cast after the effect ends.
If Blitzcrank becomes untargetable while pulling a target, the target will continue to be pulled to Blitzcrank's location. This includes Fate's Call where the target will continue to follow Blitzcrank even though he's untargetable.
However, if Rocket Grab hits after Blitzcrank becomes untargetable, the target will be pulled to the location where Blitzcrank was last targetable.
While Blitzcrank is within brush, enemy units outside of it can still be targeted and damaged by Static Field'spassive. Enemies will see the lightning bolt visual effect but Blitzcrank will not be revealed.