Can male masturbation cause genital injury?

Yes. But only if you do something that causes injury like being too aggressive or use tools that injure. Injuries range from bruises, skin tears / scratches to more serious.
No. Male masturbation does not lead to genital injury or long-term problems if masturbation if performed safely and without trauma.
Depends. As was already said...normally no, but if you add in "tools" or other objects possibly.