[Q1]
Does it sound natural? "I like watching motion pictures" instead of saying "I like watching movies"
It's not very natural. 'Motion pictures' is quite old-fashioned and technical-sounding. If you are giving a presentation about 20th century culture, for example, you might use this term, but you would never use it in casual conversation.
[Q2]
Which one is correct?
A. When the ground shook, a couple of nails came OUT OF the wall.
B. When the ground shook, a couple of nails came OUT the wall.
C. When the ground shook, a couple of nails came OFF the wall.
A is fine. B is colloquial American English, and would be ok in conversation. C is grammatically correct, but unlikely. The nails are 'in' the wall, so when the ground shook, they came 'out of' the wall. 'Out of' is the opposite of 'in'. The picture was 'on' the wall, so when the ground shook the picture came 'off' the wall. 'Off' is the opposite of 'on'.
[Q3]
What is the difference between "Keep headed south" and "Keep heading south!"?
Both are correct in American English, and have the same meaning.
[Q4]
A: How is he doing now?
B: He's hurt bad.
This is very informal colloquial American English. You might come across this in dialogue in in a book, play or movie, particularly if the character is meant to be less educated. The correct form, as you say, is 'He's badly hurt'.