Being a chippy I have worked with eucalyptus all of my life and I have made hundreds of selfbows out of many of the species of eucalyptus, most of them have been unbacked. Lots of eucalyptus species are very good for flooring, like spotted gum, red box and turpentine. The problem with the flooring you have is that it will be very dry and probably brittle so it might be safer to back it as has been suggested. Some species if eucalyptus makes fantastic selfbows and some don't, it's just a matter of trying it, and the correct limb design helps a lot as well.
The growth rings you see in the timber are just that, growth ring and not true grain as eucalyptus is even grained and you need to split it or use a grain detctor to find how the grain is running, but normally you can just plane the board flat and go from there for the back of the bow, you can sight the grain if you have a close look at it...Glenn...