Fred wrote:
[...] if you were wiring a socket, which included an earthed screw
terminal, you did not have to run a separate earth wire from the
box's earth terminal; provided that the earthed screw hole was
screwed onto a fixed tab in the box, not the moveable one. If you are
wiring something without an earth, such as a light switch, then you
have to connect to the box. Have I remembered that correctly?

There's no absolute right answer here. The requirement is that
exposed-conductive-parts must be reliably earthed. Some authorities
(e.g. the NICEIC/ESC, IIRC) are happy with the above interpretation,
others say good practice is always to include the link wire.

There's a further debate: is a flush box (sunk in plaster or drywall
construction and fitted with a plastic-fronted accessory) an
exposed-conductive-part at all? Arguably not since it's not accessible
to be touched. If you're using wiring accessories with metal front
plates though, there's definitely E-C-Ps to be earthed.

So, FWIW, my view is that it's only OK to rely on earthing via a fixing
screw to a fixed tab if the box ends up inaccessible and a plastic
accessory is fitted. With metal-faced accessories it's best to earth to
a terminal on the accessory, if there is one, and link to the back
box if necessary.

I have bought some round pin sockets to use with a table lamp and some
back boxes from Toolstation. The sockets only earth one screw hole and
guess what, it's the one above the moveable tab! The problem is that I
put the boxes in, plastered, and papered around them so I do not want
to try removing and rotating the boxes. Am I right to think that in
this case I need to run the extra earth wire from the box to the
socket?

Clearly you do.

I want to have two boxes with a switch in the top one, running to a
socket in the one below. [...]

L, switched L, and E from the ceiling rose
3-core and E for two way switching of ceiling light
twin and earth to table lamp switch
twin and earth from table lamp switch to socket

It sounds as if the first-fix wiring is all done now, but you could have
saved a cable there, assuming the ceiling and table lamps are on the
same final circuit. A 3C+E cable from the ceiling light could have been
used to bring down a neutral (grey core, blue sleeve) as well as the L &
SL wires in the 'switch drop. Then a separate feed cable for the table
lamp would have been unnecessary.

Can I comfortably fit all four earth wires into the box's earth
terminal? Or would it be advisable to crimp or otherwise connect some
of them together separately?

With 1.0 or 1.5 mm^2 T&E (1.0 mm^2 CPCs) That's unlikely to be a problem.

--
Andy