In February after the storms this year my 5 year old son spotted a piece of fossil on the beach in Herne Bay, in Kent where we live (see attached pictures - 065, 066). An early elephant tusk was found near the same patch of beach last year.... I was hopeful that it might be a piece of tusk or horn. The same day my daughter (7) found the circular indentations in pieces of flint. We've been told that they are probably moulds of chalk fossils of a type of sea urchins shell. Can you confirm this?
The little round marks in the flint are, indeed, pieces of sea urchin (echinoid). Sea urchins have spines, which can be moved in unison, enabling them to 'walk'. Therefore, the spines need to articulate, and they do so with ball-and-socket joints. What you see are the plates that include the ball part of those joints. They look inside-out because they are not the things themselves; they are casts of them.
The other thing:
I'm not good with recent mammal remains.
The inside side reminds me a little of a barnacle plate; see