Yes. Something connected to AREF. At a minimum, a 0.1 uF capacitor.
I would add this even if you don't plan to use it because you might change your mind or use it for a different project and its not like its that much more work.
Could I just solder the components straight to the chip socket as shown in the picture.
I suppose you could but I don't think I would suggest that. It would work but considering that you have to pull that chip out of the socket a lot it will probably break quickly. My general experience with sockets and ICs is that it takes a bit of finagling to get it out and without something to hold onto it could be very difficult. I would just use a really small piece (like an inch wide) of proto-board. I would be much stronger and probably easier to recreate.
BTW Thanks for figuring this out, I was looking to do pretty much the exact same thing but never took the time to study the arduino schematic and get rid of the non essential stuff.
Also I don't know if they make such a thing but an IC socket with double long posts might be very useful. I'm thinking that if you had the posts sticking down from the bottom a bit you could very easily just plug the whole thing into an uno socket without taking the atmega out of your socket. If you reprogram things and test new code as much as I do that could be nice just to save time. But I'm not sure if you would get any weird hardware interactions between the uno board and the stuff you soldered onto you board with this method. Maybe someone else can say if that would work or not.