The actual rules are pretty vague regarding how the cable should be routed. It really needs to armoured (swa) and it should be buried at a depth it won’t get damaged (around 600mm) with warning tape above. In reality you will get away with doing a simpler job if you can route it safely. For example tucked into a flower bed there’s a good chance of putting a spade through it, round the back of a shed less so. For a short run above ground might be OK.
Cable needs to be sized for the load and distance so there isn’t too much voltage drop. There are tables for this. I suspect 2.5 will be fine but if you are laying swa the cost of thicker cable won’t add much cost.
Once in the shed up or down is fine. Just needs to be safe. Hidden cable in safe zones and exposed cable protected in conduit. Metal sockets and back boxes need to be earthed properly.
As for regs different rules for different parts of the uk but its generally non notifiable unless you are adding a new circuit to your consumer unit (which you should IMO but there are other ways). Definitely needs to be rcd protected and the circuit should be tested.