Rock climbers wear smaller shoes because it gives them better support for their feet while climbing. Wearing smaller shoes allows you to be more precise with your feet, stand on smaller foot holds, and exert much more force from your toes compared to wearing larger shoes.
Climbing shoes are also designed to be very minimalistic in terms of materials and "fluff". Meaning, they have very little rubber around them in comparison to everyday sport shoes. This allows your toes to actually feel what you are standing on, and, in a way, to grip the holds with your toes. This minimalistic approach also makes the shoes seem much smaller than they are when comparing them with street shoes or general sport shoes.
It is important to note, though, that beginner climbers should probably not try and downsize their climbing shoes too much, or at all. Wearing smaller climbing shoes, which is known as downsizing, provides extra benefits in performance, but, these benefits are not really noticeable for beginners, since the routes beginners climb do not actually require or utilize these performance boosts.
Yes, but you won't gain much support from non-climbing shoes. Climbing shoes are designed to give you the maximum amount of support for your feet while climbing, even the cheapest beginner shoes.
Regular everyday shoes are over-stuffed with rubber to the point where you wont even feel what you are standing on. This will make climbing very difficult later on. For beginners climbing very beginner-friendly routes, though, it may not make a huge difference since they probably wouldn't even notice the performance gains from the shoes, they will be too focused on how uncomfortable they are. But, as you get better and advance up the ladder, you will notice the difference and understand why climbing shoes are the way they are.
