Karen,
This leaflet:
Creating Garden Ponds for Wildlife from Pond Conservation gives a lot of good advice for shallow ponds, including plants. This one from Natural England is also good:
Garden Ponds And Boggy Areas.
For very shallow water (1-2" or so) try Water Forget-me-not and/or Brooklime - I don't think you can go too far wrong with these.
Some of the other marginals such as Watercress, Fool's Watercress and Lesser Water Parsnip might also be worth trying, but they can be quite vigorous so probably too big for the small pond. Although, if they're just planted in sand then they should grow slower and (hopefully) be less troublesome.
Water Starwort and Hornwort might do well in the deeper bit.
Bob,
You could well be right about the metal bathtub - I can imagine that just sucking the heat right out of the water and causing it to freeze quite thickly. There are pond or cattle trough heaters you can get for winter use - not to warm it up so much as just to keep some of the ice off, which might be worth looking into (not sure how much they cost though).
With a shallow pond, it's not just the maximum depth but the opportunity to incorporate a range of depths especially round the margins. In a small pond, it is obviously difficult to balance having a gentle gradient from the edge with it not being so shallow that it just dries out too soon. Look through the leaflets I linmked to above.