Hi Eliza,
I think removing the slabs is the right way to go. Invariably they will be sitting on sand/grit/gravel, all excellent drainage materials.
As far as the Tomatoes are concerned I would go for deeper, 9″ minimum but 12″ preferably. The 6″ depth is really only 5″ if you allow a 1″ gap between top of soil and top of the beds for watering (you don’t want the water to run off with soil onto the surrounding paving) or a mulch for instance. This is little more than the depth of a grow bag. Whilst that depth is adequate for Tomatoes you will invariably be doing a lot of watering as it will dry out quickly. Deeper beds dry out slower but also give roots more room and helps anchor the plants better (they will still need additional support if they are cordon cvrs and indeed some bush types need this additional support also).
Beans will be fine with the 6″ depth. I can’t comment on the Cucumber or Courgettes as not liking them myself I have never bothered to grow them.
Don.
