Temperament is an extremely tricky thing affected by genetics, environment (even when they're still in the womb, to a degree), and life experiences. You also need to consider that temperament will change with age - for example, most dogs are going to be pushy, energetic, and obnoxious as puppies, even if they grow up to be amazing with other dogs. I would not expect any puppy to know how to be respectful and exercise restraint around a senior dog (or person) from day one. Manners and impulse control are things that come with time and maturity - puppy brains are literally not developed enough to have that much control, just like human babies and toddlers. There will likely be a period where the puppy needs to be supervised when interacting with your older dog and possibly parents, and should be confined or separated when you cannot supervise.
(EDIT: I just realized that your 10yr old boy is probably a human child, not an older dog, whoops. Same thing applies though - baby and adolescent puppies are obnoxious and take time to learn manners, whatever their genetic temperament might be)
Health issues have been studied much more thoroughly in purebred lines, and are to some extent more predictable. For this reason, any reputable breeder or club for purebreds is going to have information on what genetic conditions exist within the breed, even if they're uncommon and/or have effective tests that breeders can use to guarantee the puppies won't be affected by the illness.
When you get a mutt of unknown heritage, it's entirely possible that they still have the genetic makeup for any of the same issues purebreds face, but it's impossible to guess which. Now if your mutts are from street dogs you're probably less likely to see really acute problems - genetic conditions that, when untreated, kill young dogs or puppies before they can reproduce - but they can still have conditions that manifest later in life, or have a smaller impact on their immediate survival. Things like hip dysplasia (can be caused by the shape of the hip joints), a genetic predisposition to cancer, hypothyroid, arthritis, etc. affect just as many mixed breed dogs as purebreds, but mixed breed dogs (usually) don't have entire communities testing every dog's genetics and x-raying their hips and reporting every incident of a particular illness so it can be recorded in a database for future study somewhere. Many purebreds do. This can create the illusion of purebreds being genetically weaker, when really they just have more data collected and research done about their genetic issues.
That being said, there are absolutely some breeds that are in a lot of trouble due to genetic illness or conformation. Almost all cocker spaniels die of an inherited heart condition - and many Doberman pinchers, even though the breeds seem nothing alike. American goldens have an extremely high rate of cancer as young dogs. Many short-faced dog breeds like the pug or English bulldog need surgery young to be able to breath properly.
Because of all this, when you're doing your research, it's often more important to understand how common a genetic health problem occurs in the breed, how severe it is, and whether it can be prevented or significantly reduced by appropriate testing of the parents prior to breeding than just how many there are. For example, my youngest is a rare breed, and there's a disease that's actually named after that breed because that's where it was first and most commonly found. Sounds scary, right? Except that the breed community immediately mobilized and researched the condition thoroughly, and now there's a test that will easily tell a breeder if two parents could produce a puppy with that disease. Every reputable breeder does this test, and therefore it's virtually impossible to get a puppy that has this illness from one of them. That's why it's so, so important to find a breeder who cares deeply about health and performs all the genetic testing on their breeding stock, so they know how to match parents so they have the best possible chance of producing healthy puppies. There's always a risk, sure, but in many cases for breeds who do not have serious, widespread health issue, there's a lot breeders can do to reduce that risk.
As for your question about the reliability of temperaments overall... there's some similarities to my comments about health. Reputable breeders have a solid picture of the temperament they want to produce, and they do as much as they can through careful pairing of the parents of each litter to maintain or improve that temperament. But yes, any two breeders can have a different interpretation of what a breed's temperament 'should' be, or what their preference and goal for temperament is. This is why you can have a single breed - let's say a lab - with dogs who can be laid-back family companions, high-energy hunters, extremely effective guide dogs for the blind, or drug search dogs for the police. You can't get a single dog who can do all those jobs - you probably can't even get that range out of a single litter. But you have breeders who are focusing on specific kinds of labs - companionship, retrieving drive and energy for hunting, extreme stability and drive to work with people for service dogs, etc - and so you get a range of temperaments while the dogs are still all considered 'Labrador retrievers'.
So you need to be looking not just for a breed that fits the bill for what you want, but also a breeder whose temperament breeding goals aligns with your own. You might decide that a German Shepherd is good for you, but if you go to someone breeding extremely intense, high-drive dogs for police work or sports, you may find that you get an animal that doesn't learn how to settle and be calm and gentle with your older dog and parents - at least not until they're several years old. However, a breeder focusing on a stable, sound German Shepherd that is more of an 'all rounder' - eg can be happy in a lot of different households and roles - might be perfect! I'm not a German Shepherd person so I'm not saying that's necessarily the best fit for your household, just trying to illustrate that breeders of the same breed can have very different goals and produce quite different animals. Meeting as many of the breeder's adult breeding stock and dogs they've bred and now live in other family's homes is a great way to judge whether that breeder's line will work for you.