I have since found a posting on Laikas MedLibLog about this, and Laika has obviously looked at this properly! So, I can now report that you can move the search history box so it is under the search box - drag it using the grey rectangle in the top right hand corner of the search history. You can also close either or both results managers - although they will both be open in any new Ovid session.
Laikas posting is here (in English and ook in Nederlands) and is gratefully acknowledged. She talks about other things besides, so please read her posting for more!
I also discover that you can customise the limit tick boxes that appear on the main search screen. But those choices seem to disappear in any new session. I have not investigated whether it remembers the choices if you are logged into your personal account.