How does the double strike mechanic interact with other mechanics like lifelink or trample?
I think the answer to this is actually not as complicated as first appears.  (I'm assuming you  aren't confused about what happens when a trampling, lifelink creature without double strike gets blocked.)  It all hinges off the following simple (ish) rule for resolving trample:


  702.18c If an attacking creature with trample is blocked, but there are no creatures blocking it when damage is assigned, all its damage
  is assigned to the player or planeswalker it's attacking.


Recall that a double striker deals damage in both the first strike combat damage step, and the normal combat damage step.  Now, it may very well be that the blocking creature was dealt lethal damage in the first strike combat damage (and some trample damage may additionally have carried over to the defending player).  In this case,  the blocker is in the graveyard by the time of the normal combat damage step, but, crucially, this doesn't mean that this damage step doesn't happen.

If the double striker didn't have trample, then the normal combat damage step would be basically irrelevant: the double striker still counts as having been blocked, and blocked creatures deal no damage to the defending player. ETA: Additionally, the dead blocker is no longer in combat to take more damage (which may be important if the attacking creature had lifelink). However, tramplers can deal damage to defending players; in fact, any amount of damage that isn't "soaked up" by a defending creature.  Which in this case is all the damage dealt in the normal combat damage step, as the defending creature isn't around to soak up anything.  As suggested by Rule 702.18c.

The lifelink is a bit of a side issue, I feel: it doesn't change anything in regards to the double strike or trample.   It just gives an equal amount of life to the creature's controller to the damage that creature deals, in either combat damage step, and whether that damage is to a blocking creature or a defending player!

So, yes, your interpretation is ultimately correct, I'd say!