M03 turns clockwise seen from where?
From the spindle looking toward the part, that is, from behind the tool. If you stand in front of the mill and look up, you see it the other way around, and half the confusion comes from there. With that reference, M03 makes any right-hand tool cut, which is practically everything in a shop.
When do you need M04?
With left-hand tooling: left-hand taps, left-hand cutters or drills and some deburring tools that work by pulling. On a lathe, M04 shows up routinely depending on the turret and insert combination. In rigid tapping with G84 you do not write it: the cycle reverses the rotation on its own.
Can I put the S and the M03 in the same block?
Yes, and it is the usual thing: S2400 M03. You can also write the S in an earlier block, because it is modal. What is worth doing is putting the M03 in the positioning block so the spindle picks up speed while the axes move and you do not lose seconds on every tool.
Is M05 compulsory before the change or the M30?
The control does not demand it and many machines stop the spindle themselves at the change and at the end of the program. But that depends on the builder and on the setup, so writing it is the safe habit: it costs one block and works the same on every machine in the shop.
Why does M03 look like it turns backwards on a lathe?
Because the reference is still the spindle axis looking toward the part, and on a lathe the operator is looking from the opposite side. On top of that, with a rear turret the tool attacks from above and behind. What you have to check is not the visual impression but whether the edge of the insert bites the material the right way.