How does your camera change focus points unless it's on some auto setting?
If you manually set the focus point your camera won't change focus points.
Probably wrong choice of words. Not sure how the OP's model works or the features it may have but on my K3 II, I have three focus tracking modes, one of which will lock focus, and two which will adjust for movement. Even if the focus tracking mode is set to lock focus, it will still "refocus" every time you half push the shutter to recompose (unless you use the AF lock button, which will hold it for a few seconds). Since the camera is going to focus on the point you place it on, any movement in between could set a new focus point (unless you've used the AF lock) while you're recomposing.
Both my Live View (Screen) and Viewfinder have the option of utilizing all or a combination of up to 27 focus points, from single point to multiple points, to selective points. Not sure of other brands, but when using the Live View Screen, mine uses CDAF (contrast detection), low light or those situations having little contrast can cause a lot of hunting even with a short throw lens. One feature that I like when using Live View is the "Focus Peaking" option, which outlines the edges of objects in focus with white.
The view finder on the other hand uses phase detection to focus. The main advantage is the ability to focus in low light situations. The disadvantage is it's really hard to get an accurate focus in manual, especially for those like me with vision problems, something the old split prism would help.
My answer to the OP's question on using AF or manual, would be to read your manual and understand how your camera operates, because frankly there isn't a one answer fits all. The modern digital cameras (even the lower end models) have a lot of options. Knowing how those work will allow you to utilize the best options for the shot.