Does climbing put extra stress on the chain, making it stretch out faster / higher break risk? I remember taking it for granted, but then somebody gave me some reasons on climbing not affecting the chain, and I forgot about it.

Today I just got scared that my chain might break when struggling on some really steep hill I recently discovered.
The amount of force you put on the chain makes it stretch faster and have a higher risk of braking. Climbing can sometimes mean more force, but it doesn't have to. If you have the right gearing combinations on your bike, you should be able to climb without putting an excessive amount of force on your drivetrain. If, however, you have a single speed or fixed gear with a high gear ratio, you can put quite a bit of stress on the chain. If you're up out of your seat and putting a lot of power into it using your full body including arms and legs, there could be a possibility of the chain breaking spontaneously. If you regularly check your chain for wear and replace as needed, you shouldn't have to worry too much about your chain breaking unexpectedly.