mjhjmtu

The ISS has an orbital velocity of ~28000 km/h; the velocity $v$ relative to the landing site of the descent module is probably even higher than that most of the time. Once the occupants have landed, their velocity relative to the landing site is zero.

My first question is: what is it that eliminates the velocity between detaching from the ISS and arrival? Three things come to mind:

1. the spacecraft (Soyuz) engine,


2. 
   

   the atmosphere:

   
   
   

   a. descent module only,

   
   
   

   b. descent module with parachutes deployed,

   
   


3. the earth itself (final impact).

Anything else?

My second question is: how much does each of these modes contribute (measured in $Delta v/v$ or $(Delta v/v)^2$)? For the sake of the occupants' prolonged joy in space travel, I figure 3. has the smallest impact (pun intended), but how do the others relate to each other?

Nearly all the velocity is cancelled by atmospheric deceleration of the descent module, before its parachutes are deployed.

ISS orbital velocity is around 7700 m/s. An initial retro-burn of the Soyuz engines, of something like 115 m/s magnitude, is sufficient to lower the perigee of orbit into the uppermost part of the atmosphere. The orbital module and service module are then separated from the descent module. Once the descent module starts to enter the atmosphere, air resistance slows it, which further lowers the orbit, bringing the capsule into denser atmosphere, which slows it further, and so on.

The Soyuz parachutes deploy starting at ~240 m/s (first drogue chutes to bring the capsule down to ~90 m/s then the mains to reach a 6 m/s descent rate). Just before touchdown, small solid rockets are fired for the final deceleration, producing another 3 m/s of ∆v.

Thus, of the 7700m/s initial velocity, only about 360 m/s is cancelled via parachutes, reentry burn, and final retrorockets; 7340m/s (95%) of the deceleration is done by the descent module moving through the atmosphere.

(I shamelessly stole correct figures from Steve Linton's answer.)

This breakdown applies generally for all crewed spacecraft, though American capsules didn't have the final braking rockets, and the space shuttle touched down at ~100 m/s horizontal velocity, without deploying parachutes in the air; atmospheric deceleration of the airframe does almost all of the work, because it's "free" apart from the heat shielding.

The process is described here, which answers nearly all of your question. The reentry burn removes about 120 m/s of velocity from the capsule (that's your 1) and the final impact is 15 miles per hour (about 6 m/s). That's your 3. That leaves about 7.5 km/s for part 2. The only remaining question is the split between 2a and 2b, ie the velocity when the parachute opens. This source gives. Firstly 2b is given as 240 m/s, leaving 7.25 km/s for 2a. Finally, a set of small rockets fire just before landing and reduce the 6 m/s to 3 m/s.