I understand that I will lose AF after f/5.6(?) with this body,
The maximum aperture of the lens is what your camera uses when it's autofocusing. Increase the focal length without increasing the aperture opening, and the f-number increases (i.e., f-number = focal length / aperture diameter). Double the focal length, add two stops. So your 70-200 becomes a 140-400 f/8 lens.
At f/8, the camera doesn't have enough light to "see" by to focus.
but does that mean that I need to use my hand for manual focusing?
Mostly yes, and possibly no.
If you get a non-reporting teleconverter (i.e., a non-Canon 3rd-party that doesn't show up in the EXIF), the camera will still think it's attached to a 70-200/4 and will attempt to autofocus, vs. knowing it's attached to a 14-400 f/8 and not even try.
But, it still probably doesn't have enough light to properly autofocus by, so it may hunt, chatter, or have a really hard time locking on target. I used a 3rd-party 1.4x converter with my 400/5.6L USM on a 350D, and ran into this about one time in five (I live in Southern California where sunshine is copious). So, no matter what you do, AF is, at best, compromised.
There are reasons to save up the pennies and just get a 400mm lens. Using a TC never really equates to having the longer lens.
I normally never use all the AF points. I use the central AF point, half-press the shutter button, then move my camera to where I want, keeping my focus locked and taking the picture. Will this way of focusing still be possible, or does this too count as AF?
Unless you're turning the focus ring on the lens, it's AF.