Your first question is likely to open up quite a can of worms here. Is the dog a female or a male? The risks of not spaying are much higher than the risks of not neutering. First, you have unwanted litters, second you have pyrometra, which is a life threatening uterine infection. Most vets recommend spaying prior to the first heat, so usually around 5-6 months. Neutering does prevent testicular cancer and can prevent some behavioral issues when done early enough.
There is some evidence that sex hormones have influence over growth, so altered dogs are smaller than dogs altered as puppies. There has also been some evidence that the risk of bone cancer is somewhat higher in dogs that were altered as puppies, though that seems to be magnified in breeds with already higher rates of osteosarcomas.
Keep in mind, dogs will do anything to mate. Anything. If you aren't prepared to prevent your male from smashing through a wooden fence, then running a block on a broken shoulder and hopping over a 6' fence on a broken shoulder, don't keep your male intact. If you aren't prepared to keep males that determined away from your female, don't leave her intact.
As to what you should expect, most dogs are groggy and sick feeling after the surgery due to the anesthetic. Otherwise, they shouldn't be in too much pain, though, yes, some dogs need a cone to prevent them from licking or chewing at the incision. You can also use a t-shirt to cover the area, really anything that prevents them from being able to lick or chew at it. With a spay, the incision is larger and eventually fades away. With a neuter, the scrotum is left intact and gradually shrinks until it's unnoticeable. Sometimes, the scrotum swells, which happened with my dog, but it's not a big deal. Kabota didn't seem to be in any pain and I couldn't find his scrotum after a few months.