Great questions, I wonder if these would help?
For subtraction, I always tell my students it’s a loss of a debt. In terms of the numberline, I ask students to start with the first number, -6 and subtract -4. If they move toward the left I say, that looks like you’re subtracting a positive four. Negative 4 is the opposite of positive 4 so you need to move to the right.
For multiplication, I think in terms of football yards. If a team loses 3 yards 7 times, they’ve lost a total of 21 yards.
Division…I just tell them division doesn’t exist. It’s multiplying by the reciprocal. so 8 / -4 is 8 * -1/4. same rules apply as multiplication…