@Bluephoenyx touched on the spay/neuter which can be common, but can vary. Always ask what the spay/neuter policy is; there are perfectly valid reasons to not spay/neuter or to delay it until later in life. There are some threads here which go into the details, so I'm just going to gloss over it and mention the hormonal issues such as incontinence that can happen as well as the problem with certain cancers which can be greatly exacerbated by spaying, especially if done before the dog has fully matured. It's not weird to ask if this or that dog is spayed/neutered, as the average pet owner would want to know if that's an included expense or someting they will have to pay for later. Likewise, it's not weird to just look for testicles or a spay scar if you're right there with the dog for the same reason.
I've found small operations and breed specific rescues are more likely to have and rehome an animal without modification, especially if the person adopting is knowledgeable on dogs and preferably the specific breed, but it's going to vary wildly from place to place.
I have adopted a few times when I've encountered a dog who has really stood out and made an inpression on me. I would like to make an aside here and comment that you should never make sex a primary reason for getting a dog, my last adopted dog came to me intact and I had no intention of altering that, but health problems override desire and after seeing several sspecialists for the problem, he was eventually neutered. He's still every bit of that special dog who won me over because sex was not a motivating factor in adopting him, it was merely a bonus.
There are some really fantastic dogs that can be found at rescues/shelters, you just have to be selective and pick based on the characteristics you want and not by emotion. Never pick a dog because you feel sorry for it, pick one who's the best fit and interacts with you the best.
I tend to choose a puppy for a variety of reasons, but these are foremost.
First off, I like to know what I'm getting so I tend to research the breed and line, if the breeder doesn't have tests for all the usual health problems and the bitch and sire don't have great results, I keep looking. As the saying goes, the apple doesn't fall far from the tree so what you see in the parents is typically a good indicatior of what the pup will be like as an adult. Yes, it makes the puppy I choose more expensive up front, but with careful selection it can save a fortune in heartbreak and vet bills later.
Secondly, I expect more from my dogs than for them to be happy companions to follow me around or relax with me on the couch, I train my dogs to work with me and I expect a lot more out of my dogs, so I want to ensure the dog I'm investing the next 10-15 years into will be capable of what I need. Puppies learn faster and exhibit more plasticity in learning than adults, so if that maters it is prudent to select carefully if considering an adult.
That said, some people just don't have the time or resources to manage a puppy and get in the socialization and training necessary.
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