Yes on the shelter process but it varies to much between em. I know many shelters try to evaluate and then write up a dogs history, others do little to nothing. I don't know if this is driven by volume, legal reasons, other.
Before I got Cooper, I had been watching some Dog Whisperer shows. Say what you want about him, he had a great episode on how to pick a dog at the shelter. Armed with that, I took a few dogs on walks. Cooper (or Copper, as he was then called) was much better behaved than other dogs. He didn't pull much. He seemed friendly and receptive to feedback. He was 10 months old, spent two of those in the shelter system, and was just an easy, easy dog. Everyone he met thought he was special, unique, "boy you got lucky." My parents and in-laws just adored him, even though my MIL was actually scared of dogs.
I picked Loki based on a picture I saw online, five days after my cat died. I was desperate for another ball of fur and wanted a high energy ball of fur. Loki also had kennel cough when I got him (this was fully disclosed; he was quarantined, which is why I didn't find him when I looked around, and probably why he hadn't already been adopted). So he was probably not as rambunctious as he normally would be. I love Loki dearly and he does not have terrible manners, but it wouldn't be hard to draw a line between my approach and the outcome. (He also suffered from second child syndrome - it's just a lot harder to work with two dogs.)
Two data points aren't much to work with, but I think taking a dog for a walk can tell you a lot about its temperament.