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.

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.

Take care of them and take care of your feet.
Make sure you let your shoes dry after climbing sessions, and don't put them in your bag. Also make sure your feet are clean before putting them on.

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.

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.

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.