6mm cable for maybe 40 feet of circuit?
I’d say 2.5mm would be fine – what is the expected load? – typically the biggest current drawer would be a lawnmower at 1.5kW, so no need for a bigger cable, for either current consumption, or voltage drop.
No, you would not need any ducting if you use armoured (SWA) cable. to comply with the Regs, it will have to be RCD protected, either at the shed end, or from the consumer unit, entirely up to you how that is done. It does need Part P certification. This is regularly ignored, with no real penalty if you do DIY.
The cable should be buried to a depth that it is not going to be damaged in normal use i.e., at least deeper than a spade if it crosses a garden, preferably 600mm deep. If under a footpath/slabs, then a few inches would be sufficient, as it is protected from mechanical damage by the slabs.It should have a warning tape laid above it.
Price – MCB, or RCBO in CU, £3 – 20, cable ~ £1.50/m , CU in shed £20,fittings in shed £20-50, labour, £100ish, probably 3 hours work, without digging the trench, so around £250 I reckon.
If you’d like any more info, then please let me know.