Philip,
The normal ILS procedure for the 747-400 (and I presume a number of Boeing types) directs for Gear Down when the glideslope pointer begins to 'float' (starts moving downwards) on the pilot's nav display, with final landing flap selection on glideslope intercept (usually around 3000ft AGL, depending on the vector given). This makes for a slow and noisy approach, however, and is only used in inclement weather, or when there's a requirement to configure early.
99% of the time, a 'Noise Abatement Approach' ILS procedure is flown, in which Gear Down occurs at 2000ft AGL, followed by landing flap selection. In the case of 16R, this should occur 6.2nm from the threshold (you can use Google Earth to see where this is exactly). However, there are often directions from the radar controller to maintain a certain speed (eg 160kts) until x miles out, in which case the selection of gear might be delayed beyond 2000ft. Similarly, a direction to reduce speed earlier than usual would trigger the taking of gear and landing flap earlier than usual. As such, the point over the ground at which gear down occurs is fairly variable. As you mentioned, the gear is also remarkably good at fixing an approach that becomes a little 'hot', and might be taken earlier for that reason
Depending on the operator, the approach should be 'stable' (and in the full landing configuration) by 1000ft AGL.
Hope this helps.
Will
Ps. The normal 3 degree glidepath is approx. 330' per nm, so divide the above heights by 330 to see roughly how many miles from the threshold they are.