Hi Bea,
I think that’s the hardest part of setting out a new lawn area – getting a nice even surface and it pays to take your time over this aspect. After a year of being covered all weeds will be dead and you will have a nice clean surface to work with. Once your happy that the area is level or contoured as you require just lay your sods.
As for the flower beds I can attest that the no dig approach made a significant difference for me. The flowers/shrubs in my herbaceous borders were bigger, stronger and the colours seemed to me more vivid. I’m still amazed that simply spreading compost on the surface could make such a difference but it really does.
One final thing. I think you should apply the compost to the bed areas in October/November which will give plenty of time for the worms to incorporate it into the soil over winter, and of course give the frost time to break up any lumps. The beds will then be in perfect shape for planting into in spring.
Don