Yes, climbing shoes can get wet. In fact, many people wash them to clean them, some even put them in washing machines too (I have never tried it, there is quite a bit of debate on the washing machine approach).
Also, while climbing, even if it's not really noticed, people sweat in their shoes, especially in the summer and when climbing outdoors. It's very common to take off climbing shoes to find out that the insides of yours shoes are pretty wet.
Everyone has a different limit when it comes to exercising, including climbing.
When I started, I climbed about 4 times a week for about 4-5 hours each session. At that time, I didn't feel that it was too much since much of that time was spent also resting and planning, looking back now though, I realize that it was way too much, to the point where I injured myself without even knowing it.
Knowing what I know now, if I were a beginner climber, I would climb just 3 times a week, and just for 2-3 hours. I would work on learning how to climb, and building up my strength, fingers, and all the muscles that were not used to climbing and to the different positions and strains it puts on your body.
Climbing shoes normally do not shrink, they actually stretch and get looser.
The amount at which they stretch depends on the material the shoe is made of, and of course, on how often the shoe is used. If climbing shoes are rarely used, or used not frequently, they will take much longer to actually stretch.
The material the climbing shoes are made of plays an important role in how much the shoes actually stretch. Leather shoes tend to stretch quite a bit, which is part of the reason why many climbers buy shoes that are smaller than their actual climbing shoe size, since they know that after some time, the shoes will stretch and will be a good fit.
Climbing shoes that are synthetic and not leather tend to either not stretch at all, or stretch very little. I have quite a few pairs of non-leather climbing shoes, and I cannot say that I have noticed any stretching in the shoes, and I climb 3-4 times a week. They still feel as tight as the day I bought them, just used.
As someone who is not flexible myself, but has flexible friends, I can say that it is very important, not just for finishing routes, but also for preventing injuries.
There have been quite a few routes that I have not been able to do, only to see my friends do them using some sort of crazy body folding beta. Lifting their legs up to the level of their ears, and other moves that I had no chance in doing.
On top of that, certain routes require a certain level of flexibility, and not having that flexibility, I felt close to injuring myself quite a few times while trying the moves.
In general, though, I think flexibility can help even on routes that don't require you to be so flexible. It can allow you to do them more statically and be in control better than if you were not flexible.
I have started training and working on flexibility, and I do see quite an improvement while climbing, though I am still a long journey away from declaring myself as "flexible".
There are sprays that are used for shoes in general to help reduce/prevent bad smells. Normally, I would spray my shoes after each session. Realistically, though, I forget, so I spray them once a week.
Another method I found that really helps with the smell, but not only that, makes the shoes actually smell good, and makes their insides softer, is putting baby powder in them. There are some powders that are scented, and some that are even made specifically for shoes (which probably won't count as baby powder, but same concept).
I usually put it in my shoes once a week (If I forget then once every 2 weeks), and it usually helps eliminate the bad smells almost completely, especially when I spray the deodorant in them.
Another crucial thing is to air them out. Don't put them in any closed place, not in your bag and not in your car or trunk. I learned from this mistake the worst way. Putting your shoes that are already smelly in a closed place just amplifies the horrible smell. I usually have my shoes clipped to the back of my bag, never closed in anything. And I never leave them in the car, as this again is a closed space. Doing this will cause your shoes to smell bad, but also your car, especially in hotter times like summer.
The logic is that tighter shoes gives you more precision on smaller holds, and helps keep the shoe in place when doing things such as heel hooking and general heel usage.
If your heel moves in your shoe while heel hooking, that could be an indication that your shoe is either too big, or that it doesn't match the shape of your foot. Either way, it can, at some point in your climbing journey, reduce efficiency.
Same with being precise on smaller foot holds. If your shoes are not tight enough and are loose, you may not be able to stand on certain foot holds, specifically those very small and easy-to-slip-off-of foot holds.
This is why climbers usually downsize when choosing climbing shoes.
With all that said, though, I feel it is very important to mention and to keep in mind that the level of precision you would need from the climbing shoes depends on the level of climber you are. Just because you have professional, expensive, and tiny climbing shoes does not mean you will become a professional climber. If you have just start out climbing, there is no point in downsizing. Even if you have been climbing for a year or two, downsizing may not really be required. It may not even be dependent on the amount of time you have been climbing for.
Usually, for boulders for example, routes up to v7-v8 don't really required such precision. Many times you can even see more advanced climbers climbing v7 and v8 and higher even without shoes, so even the climbing level as an indication should be taken rather lightly.
My belief is that when you will need tighter climbing shoes, you will know it, because you won't be able to climb that specific route because your shoes will just be too loose. Until then, especially if you have been climbing for under a year or two, very tight climbing shoes are really just not needed. In my opinion, of course.
Another reason for not downsizing when you are just starting out: It hurts!
Climbing shoes are not comfortable. They can be more comfortable in comparison to other climbing shoes, but as a shoe in general, they are just not comfortable, and can even hurt. Even more than that, if you try to jam your feet into small and tight climbing shoes without properly conditioning your feet to gradually get used to that kind of pressure, you could end up injuring your self in some way.
Also, as mentioned above, there will be no real benefit to you if you use professional and super tight climbing shoes as a beginner compared to using beginner climbing shoes since the routes you will be climbing probably won't include holds where you would need that level of precision, and the tight professional climbing shoes would have no added benefit in comparison to looser, cheaper, beginner level shoes.
One more thing, just to point out that some people, including myself, actually go up in size from their street shoe size. I go up about 1 size from my street shoe. Going down even half a size is just too tight, too painful, and just hurts my climbing rather than improving it.
I usually climb alone due to my working hours, but later on during the week I discuss the new routes in the gym with friends, and usually what happens is that I find out that I had no idea what the intended beta was, and that I completely broke the intended beta. Sometimes I make it easier, other times I make it much more difficult than it's supposed to be.
Usually my friends are in the gym when the routes are being set, so they know the beta, but I am not, so I almost never know the intended beta, and since I climb alone, I usually make it up as I climb. So, it happens pretty often.
Climbing on the moon board really helped with my sit starts, something that I was fairly bad at. I just couldn't get myself off the ground.
After dedicating a few months to climbing on the moon board, besides getting stronger in my core and grip strength, sit starts became much easier for me, since, at least in my gym, the moon board sit starts are much harder than the regular routes sit starts.
Besides that, I believe that supplementing climbing with weight training can help a lot with it.
I know some climbers that have been climbing for quite a few years, and while I do not know how their feet were before they started climbing, I can say that their toes do have a shape resembling that of a climbing shoe. Mainly, the toenails have a curve to them, and their toes are sort of squished together.
Though, this doesn't mean it's necessarily because of the climbing shoes. I know people who have similar feet/toes that do not climb, and I also know climbers whose feet have not changed, so it's hard to say for sure if climbing shoes change the shape of your feet or toes.
I have been climbing for a few years now, and my toes seem to be more curved now than they used to be. The tips of my toes used to all be round, now they are sort of "pinched" in the corners.
It took me 6 months to do my first v6. I think I remember every part of that route (it was almost 3 years ago), and the feeling I had when I was about to match the top hold.
My goal was to go up a grade every month. That stopped at that v6, though.
Since then, I have moved to a different gym, and in this gym, the grades are much softer. There are new climbers that can do many of the v6 routes, and they have been climbing for just a few months. So it really depends on the route setters, and if they are soft grades or not.