Hello Rowan,
1. The altitude at which the flaps will be retracted can depend on a few factors. The aircraft will commonly climb to about 1000' above airfield elevation and then transition to a shallower climb as the flaps are retracted through all the stages to clean. 1000' is a typical altitude to start this process, however it may be delayed higher for other factors such as clearing a close in height requirement (obstacle or airspace) or for noise abatement.
2. It depends on what the problem actually was, and when the jet detected the problem. As soon as the 747-400 detects a problem with flaps, an appropriate message in amber will appear on the centre instrument panel (on the same "TV screen" as the engine instruments). There is a hierarchy of messages from MASTER WARNINGS (bad) to simple memos. A flap issue will generate a yellow alert message with a brief aural tone to get the pilot's attention. It would be accompanied by a MASTER CAUTION light. There are a number of possible problems the aircraft you were on had; from your description I would hazard a guess it was a FLAPS DRIVE problem. This is where one section of the flaps are mechanically stuck and the primary and secondary means of retracting/extending will not work. The rear flaps are divided into four sections and to preserve controllability, are linked symmetrically and will fail sympathetically, e.g. if the left inboard section locked, the right inboard section would lock as well. On climb out this would mean you will have that section stuck where it failed (symmetrically), the others sections work normally. You will dump fuel to reduce weight and therefore landing speed, set the flaps at their maximum landing setting (FLAP 30) and land. As you have less flap out, the checklist will direct the pilots to fly an approach speed that corresponds to FLAP25 approach speed + 20 knots......call it 25 knots faster than normal.
3. The tires bursting is a little unusual, even for a FLAPS DRIVE landing. Although landing faster, its not a huge deal and at maximum landing weight at Heathrow there is plenty of runway. Unless there were other circumstances it may have been over enthusiastic manual braking on the part of the handling pilot. As kuobin mentions, it may affect landing distance and can possibly further damage the aircraft, but not likely to offer any particular risk to passengers.