Nathalie, the split is due to shrinkage and is along the glue line. It could be repaired by disassembling the top, rejointing the edges and gluing back together. However, it appears there is a type of breadboard edge, and if it wasn’t applied correctly, could be the reason for the split. If you can determine if it was pinned, they can be drilled out and it should come right off. If the bb is attached with glue, that is not correct and explains the split.
All this said, looking at the other pics, I would be leaning toward not trying to fix it. Yes, I know it seems wrong maybe, but the piece looks old and has quite a bit of “character”.
If the cracks are really bothering you, they can be filled with colored wax filler.
I would not sand it unless you are planning to restain it, and that us a “no return” type procedure.
I twould just do a good cleaning, a wipe down with furniture oil or oil/wax (I really like Howard’s Feed n Wax).
Hope this helps!