I have to carve the letter C out of an 8 inch cake and having never done this would like some help. My question is would it be better to partially freeze the cake and then try to carve it or just carve it after it has cooled down from the oven. Any suggestions would be appreciated. Thank you.

I carve the general shape when it is cooled, and then freeze it. When it is hard from the freezer, then I do the more detailed shaping, and it is much more co-operativeQ

I like to refrigerate the cake until completely cold and then freeze it for about 15 minutes before carving, any longer and the filling freezes and the carving gets a bit difficult.
Of course if you are filling after you carve you can always freeze the cake completely.
It also helps if you make a cardboard template to follow so you know where to cut.
If you have a bread knife that will help also, but be careful when working with frozen cake,I undestimated it once and the knife slipped right though it and my finger, stiches and cake not a good combo