Does anyone know what the rules are for just randomly deciding to fly
over a built-up area like this for the fun of it?
In UK, the master document for air regulations is The Rules of the Air Regulations (ROTAR). UK being still part of the EU, they refer to a common set of rules at the EU level, the Standardised European Rules of the Air aka SERA (qui sera sera...)
UK ROTAR state a minimum flyover height, one for VFR, one for IFR:
The minimum heights for VFR flights shall be those specified in
SERA.5005(f) and minimum levels for IFR flights shall be those
specified in SERA.5015(b)
and SERA says:
(f) Except when necessary for take-off or landing, or except
by permission from the competent authority, a VFR flight shall
not be flown:
(1) over the congested areas of cities, towns or settlements
or over an open-air assembly of persons at a height less
than 300 m (1 000 ft) above the highest obstacle within a
radius of 600 m from the aircraft;
(2) elsewhere than as specified in (1), at a height less
than 150 m (500 ft) above the ground or water, or 150 m
(500 ft) above the highest obstacle within a radius of 150 m
(500 ft) from the aircraft.
So for VFR: 150 m, but 300 m in populated areas.
- (2) elsewhere than as specified in (1), at a level which
is at least 300 m (1 000 ft) above the highest obstacle
located within 8 km of the estimated position of the aircraft.
And for IFR: 300 m above obstacles.
As you see this is not very high. In addition, as you mentioned, aircraft landing or taking off will indeed fly below these minimums.
Jets landing at LHR with a stupidly shallow glide path angle
Landing cannot be done at high angle, the proper slope is the result of two elements:
Aircraft cannot afford a high descent speed, since the vertical rate determines how hard the landing gears will touch the runway and whether they will be damaged.
Aircraft cannot slow down under a given speed without losing their capability to create lift. This minimum speed and the previous maximum vertical speed determine a maximum slope.
This glide slope has been chosen so that it is compatible with all sorts of aircraft approach speeds and vertical rates.
A standardized value of 3° is common. But when the approach area is limited, or when this is desirable to maintain a higher altitude a longer time to reduce noise on the ground, then this angle can be increased to 3.5 or 4.5°, or even 5.5°, this is the case for some approaches at Heathrow as @TomMcW told us.
Alternatively, some airports have lower angles, down to 2.5°.