I don't know his actual connection to the sport or its origin, but, from the very few videos of him climbing and talking about it, you can see how much he loves the sport.
They are tight to help you use your feet better. That tightness is what allows you to keep your feet and weight on small holds, to pull your weight with your toes and heel, and to stand on difficult holds on terrible slabs.
The tightness allows you to just use your feet in a much more efficient way, maximizing you precision and foot power.
You can, but why would you? Climbing shoes in general are not comfortable and can be painful, walking in them usually doesn't feel any better.
Though, in some cases walking in climbing shoes can help soften them up a bit at the beginning of the session, but even that is usually very limited.
It is very common for them to hurt, but, they are not "meant" to. They are meant to be very tight with no dead space, and this causes them to be painful. They don't always have to be painful though, it depends what you are aiming for with the shoes, what kind of climbing, what kind of routes, etc.
Loose shoes are great for warm-up routes, but generally bad for hard routes. Tight shoes are great for hard routes, and can be used for warm-up routes, but if you have looser ones, you may prefer them for the warm-ups over the tight shoes.
Yes, leather climbing shoes stretch and can even become quite baggy if you choose the wrong size and they stretch a lot. Synthetic shoes barely stretch, and even if they do, its so minimal that most don't even notice a difference.
Yes, you can go bouldering alone in a climbing gym. Bouldering in climbing gym doesn't require any special safety equipment as the gyms have everything that is needed.
Bouldering outdoors, on the other hand, would require at least one more person for safety reasons. The person would be your spotter, and would be there to help in case anything goes wrong.
I have a fear of sports climbing. The moment i get a little bit too far from the ground, each clip becomes more and more nerve racking and stressful.
The more I do it though, the less stressful it becomes. After just a few sessions I get used to it and the fear goes away. But, the condition is that I do this often. If I sport climb once a month, the fear doesn't go away. If I do it at least once or twice a week, the fear goes away pretty quickly.
So what helps me control my fear is actually doing it more often.
I think this refers to the head forward and slightly hunched back posture climbers tend to have.
Of course, not all climbers have this posture, but many do, it has become a sort of stigma, giving it the name "climbers posture".
Because leaving them on in between routes can end up being painful, shortening your climbing session, and making every attempt on the route more and more painful and less efficient.
Taking your shoes off in between routes lets your feet rest a little bit.