Yes, all are doomed to reanimation in The Walking Dead, unless prior to death they destroy their brain or have someone do it. As to why so many zombies exist due to not having destroyed their own brains, it's probably because the infection spread so quickly during the onset, that people didn't have time to know what was happening. A person suffering from the most superficial bite still dies of infection in less than a day. The whole plague had wiped out the majority of the human race in only a couple months. That's fast enough that there'd be billions of zombies out there in short order.
As to the question, I'm in the group that doesn't mind either walking or running, but the truth about The Walking Dead is that they are not strictly one or the other. If you pay attention, you see both, & what I mean by that is that they move at average human speed
, not old school lumbering zombies, & not 28 Days Later lightning fast Zombies
The motivator is food. They ain't too quick witted, the geeks, so they're slow to make the connection, but once they got food in their sights, they start to put a move on right quick, with physical speeds comparable to the person they were & condition of the corpse. However, in herds they tend to move slower because of the numbers involved.
The only times they are slow individually, are when they're just out roaming & have no motivation to be moving fast, just like a regular person. One of the best things about the show is that the zombies are in all ways that matter just reanimated people, with vastly diminished brain function, who are spreading infection, & have a pretty indiscriminate craving for live flesh.