No prob, FlightPath. Sorry for the slightly cynical reply earlier but I was thinking that there might be a bit of winding up going on.
You're right that there is a matrix of 'what ifs' that a controller uses - but that's the just day to day work. What the controller is routinely doing is making sure that any separations or other rules that should be applied, are in place - a lot of the work effectively is proving that everything is as it should be. In high-density airport traffic environments, where every second or mile that two aircraft could have been closer simply adds up to greater delays for later flights, other things being equal, the controller is trying to pack the aircraft as close together down the approach or get them off the ground as quickly as possible whilst staying within the rules.
Some of the rules are absolute and inviolate, others have a variety 'options' that can be applied in particular circumstances. In some cases there are additional sets of rules that can be used which modify some of the others in the right circumstances.
One example is 'reduced separation in the vicinity of an aerodrome' - in very simple terms, if the controller (or the pilots of the aircraft involved) can see the aircraft and make sure they won't collide, many of the IFR separations can be reduced. The controller can use this procedure to separate the aircraft for as long as the right conditions pertain but after that he/she must have another type of separation in place. You mentioned one such separation where departing aircraft turn onto tracks split by 45 deg or more - but if it's a nice day and the controller can see the aircraft until they are well on their way and, say, seen by the radar controller 5 miles apart (i.e. the normal radar separation minimum for that environment), all is well and at least one of the IFR separations has been in place at all times. In some cases, a small radar in front of the tower controller can be used to 'watch' the aircraft if the weather is not good.
WT separation is one of the rules that cannot be varied (OK, in some countries, there are a few oddities but generally the rules MUST be followed. However, the risk caused by WT to another aircraft differs in different phases of flight and the separations reflect these differences, both in the distance that might actually exist between the aircraft and the way that they are stated (some in distance, others in time). You are quite correct that the extract from Doc 4444 that you have posted relates to aircraft flying close to an airport, but not actually on the runway, In fact the separations that you have posted are most usually applied between aircraft on the final approach path (usually established on an ILS or turning to intercept the LOC). They do also apply to departing aircraft as the text mentions but in most situations some other separation - or just the routes of the aircraft - mean that there is a greater distance between the aircraft anyway.
There are a lot of minor variations in WT rules around the world - most of these are local refinements of the ICAO rules based on empirical experience. A common example is the B757, which generates particularly strong wingtip vortices for its weight (the usual determining factor) on approach. A number of States have put the type into a higher category that its weight would normally require or have specified an extra mile or two of space before the next aircraft follows it.
Finally, you may come across other time-based separations for departing aircraft - usually set out on tables and referenced to the speed that the aircraft climb at. These are IFR separations and, in my experience, are used where aircraft will be following the same route on a SID for the early part of the flight and the times in the table are designed to ensure that, at the end of the SID, some other form of IFR separation will exist - it's the same basic principle in play that I mentioned earlier, there has got to be one separation in place at all times.