Would I able to take moon photos with the 2X teleconverter with manual focus with my Nikon 55–200 mm f/4–5.5 lens?
Yes, you will be able to take moon photos. As you noted you, you will have to manually focus (which you would want to do anyways when shooting the moon). You will probably be somewhat underwhelmed, though. At 400 mm on a crop sensor body, the angle of view will be 2.3° (vertical) by 3.4° (horizontal). The moon is about 0.5° across. Therefore the image frame is about 4.6 times the moon diameter (vertical) by 7 times moon diamater (horizontal).
Just taking pictures of the moon itself, 400 mm will not be very impressive. However, if you shot the moon low on the horizon, with buildings or landscape features in frame to set it off, that 400 mm can make for some very impressive shots.
You also mentioned "other planets". Without a telescope and T-mount adapter to mount the camera on, planets will only appear as dots.
See also the following questions:
How much will image quality be reduced by using a 2X teleconverter?
It will be reduced quite a bit. Teleconverters work better with longer, fixed focal length lenses, than with shorter zoom lenses. Personally, I wouldn't consider using a teleconverter on anything less than 200 mm.
Compared to a fixed 400 mm lens, the 55—200 mm with 2X teleconverter (at full reach) will suffer primarily from some softness or loss of sharpness, and will be two stops darker. Regardless, you will still have a higher resolution image of the moon than you can currently take.
I would, however, recommend a sturdy tripod and head when shooting. Even though the moon is quite bright, you will want to shoot a shutter speed no slower than 1/400 second (using the inverse-focal-length rule of thumb).