Does the winding have to be solid wire, or can I use stranded wire, or does it matter?
No difference electrically, either would work so it's a matter of preference. Depending on wire gauge and availability, for transformers I prefer PTFE insulated high strand count wire ($$) or simple enamel coated solid wire (scrap from an old motor or transformer).
And should the windings be flat wound, such as ribbon cable, or twisted together?
There are a few fundamental electrical advantages to ribbon, twisted versus flat however would you notice the difference at 10 MHz? In my opinion, ribbon and twisted wire can be a pain to form not to mention the added expense of ribbon wire. Furthermore, from what you describe it seems like the transformer configuration is somewhat arbitrary so you may need to try a few iterations before achieving the desired result. So I would suggest keeping it simple for a first go around then experiment later if you are curious.
John - AB5Q