A chalk bag is not required, neither is chalk itself, but it is very helpful, especially if the holds are a little bit "over-touched", and if you sweat from your palms.
Pulley and skin damage are probably the most common finger injuries or finger damage for climbers.
They can both normally be prevented, though, if the climber is first of all aware of these two risks and knows how to minimize them, which largely means paying attention and listening to your body.
It helps us prevent skin injuries such as tears and flappers. if the skin is rough, it can get caught on holds and could tear, which can be a nightmare for climbers.
Filing rougher and bulgy areas of the skin helps prevent this.
They "go" in a bag or some kind of container, keep it with them until they get down from the route (or up), and then throw it away when they can.
We climb and we exercise. As climbers, we are pretty active. We utilize many or most of the major muscles and smaller ones also. We also have additional non-climbing workouts such as weight lifting.
Of course this is not true for all climbers, but for many it is.
Not really. They maybe uncomfortable and borderline painful, but you shouldn't actually feel any pain.
If you have blisters or you aren't used to climbing shoes, that could change the situation. Until your blisters go away or until you get used to your shoes, you could feel some pain.