Convertible or Slate:
This is often more of a personal choice than anything else. With regard to performance concerns though it is not so much that you give performance away for a slate - the smaller convertibles have similar specs. I use a HP TC1100 and a Motion M1400 regularly. Both have centrino 1.1 - 1.2 GHz processors and I can tell you from experience that for note taking and general MS Office applications this is absolutely fine. If you want to play 3D games on it then you would be better off with a Toshiba M4 and a small trolly. Most slates have a docking station and/or a removable keyboard option. I would recommend both for the most flexible solution. Docked it is just like a desktop. With the removable KB it is just like a laptop and in slate mode it is something else all together.
Screen Size. As above I use both the TC1100 (10" screen) and the Motion M1400 (12" screen). Both run at the same resolution (1024x768) and I really don't notice much difference when using them as a slate. If portability is important to you then you get a much smaller device without giving much away going for the 10" screen. Remember that you can use an external monitor when in your room - you will only be using the smaller screen when writing on it. This means that the device has to be reasonably close to you, so the small screen size will probably not be an issue - that has been my experience anyway.