Indoor rock climbing is just called indoor rock climbing. Though, the term could refer to a few different styles of climbing.

Indoor climbing could refer to bouldering, sport climbing, top rope climbing, auto-belay climbing, and speed climbing. Each of these may have similar traits to each other, but they are each different in their own way.

The most common and popular forms of indoor climbing are bouldering and sport climbing. So it is most likely that when referring to indoor rock climbing, it is actually a reference to either indoor bouldering or indoor sport climbing.

Tall and short people both have the potential to become amazing climbers. Climbing does not depend on the climbers height.

There are times when being short can be beneficial, such is in very compressed movements and positions, and there are times when being tall and having a long arm span can be very useful, such as reaching farther holds easier.

The main thing to keep in mind is your technique, and how you take advantage of your height, be it shorter or taller.

Sport cloths, nothing special really. Depending on the gym you will be climbing at, you may have to wear climbing shoes, which are usually provided at the gym as rentals. Chalk bags are also usually provided or can be rented for that session.

Other than that, something comfortable would be best. If it will be your first time, or you are a beginner, long pants may be wise to help prevent leg scratches if you slip or hit a climbing holds. They can easily scratch the skin without you even noticing.

You can walk in climbing shoes, it is just not recommended. Climbing shoes are not meant for walking, they are meant for climbing. Walking in them can just wear the shoes out quicker. Not to mention that they are also pretty painful to just walk in.

A lot of climbers like to walk in new climbing shoes to help stretch them out and break them in, so that could be an exception to the above. After those shoes have stretched, though, it would just be wearing them out quicker.

When you start to see a hole in them, and before the hole goes through the fabric of the shoe, otherwise it will just cost more to get them resoled and to fix the hole in the fabric rather than just resoling them.

The most common place for holes is in the front tip of the shoes, where your toes are. This is usually due to either poor footwork, or just the lifespan of the rubber on the shoes. Softer shoes with soft rubber tend to wear out faster. Once you start to see a hole, you may have a few more sessions left before the hole really becomes bigger. Waiting too long though can just cause the hole to expand and eventually there will be a hole in the actual fabric of the shoe, which usually is also fixable, but just costs more.

Climbers take their shoes off to reduce the pressure on their feet. This helps them reduce pain and prolong their session.

Climbing shoes are usually very tight, and, after wearing them for a while and climbing a few routes, they can get pretty painful. Taking them off in between routes allows the climber to reduce that pain and pressure from the feet.

This is more common in climbers who downsize their shoes, which means that they take a few sizes smaller than their actual street shoe size.