Nautilus will dash all the way to the target's location if Dredge Line lands the killing blow on them.
Dredge Line will not interrupt any movement commands Nautilus was issued pre-cast so long as their directions somewhat correlate (this is intentional to smooth out traveling/escaping)
Dredge Line's dash's cast time and projectile speed can be manipulated to override most crowd control effects Nautilus is affected by (including displacements) so long as both their and Dredge Line's cast timing can be aligned.
Spell shields will block the ability (but only a single application)
Titan's Wrath has no cast time and does not interrupt Nautilus' previous orders.
The first half of Titan's Wrath's damage is dealt on-hit (so long as the shield remains) while the rest is applied over the next second (regardless of Nautilus' attack speed)
Nautilus' basic attacks use a different animation while Titan's Wrath is active.
The damage over time will always apply when this animation plays (even if the shield fades during a basic attack)
Titan's Wrath's shield can be artificially extended by using others (Eye of the Storm, Help, Pix!) in order to keep it from breaking before the maximum duration ends (they will not allow Nautilus to apply Titan's Wrath's damage if they are still active when it does)
Spell shields will block the entire ability for the primary target.
Depth Charge cannot be intercepted by Wind Wall or Unbreakable (despite being a projectile) does not circumvent terrain or units, and will accelerate over time to ensure it cannot be outran (only extreme movement speed could potentially make it so)
Moving a vast distance away (recalling, teleporting) before Depth Charge hits will make the projectile disappear.
The stun and the displacement are applied at the same time (there is no sequence/delay that would extend their total duration)