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.