Welcome!
Dogs: Boulder has complicated trail-specific rules about dogs. Everything from a ban on dogs, to leash required always, to leash required seasonally (bird nesting season), to “voice & sight” control. Basically, you will need to take your dog to Open Space Mountain Parks (that’s the name of the city dept that deals with such things) and have them certify that your dog can respond to commands and won’t bother wildlife or humans. http://www.ci.boulder.co.us/index.php?option=com_content&task=view&id=3051&Itemid=1041#tag_pgm Otherwise, you’ll need a leash on OSMP property. It’s complicated, because a lot of popular running trails, such as the Boulder Reservoir, are in a different department (Parks and Rec) and have their own dog rules. The local organization that represents dog owners in preserving access to open space is http://www.fidos.org/ They have good links and information.
As for trails that are spaz friendly, here is a map which gives you the location of things. http://www.ci.boulder.co.us/index.php?option=com_content&task=view&id=2794&Itemid=1035 A few choices: The South Boulder Creek trail heading N from Table Mesa is dead flat and wide (you could ride a road bike on 99% of it)(also, it’s better to access this trail from Baseline). Going S from Table Mesa is also nice, but no dogs. A few miles south, the Community Ditch, Greenbelt Plateau and High Plains (brand new) trails are very easy. They have a variety of trailheads. Dry Creek is flat and dog friendly, but very short. The Teller Farm trail S of Valmont is very easy and dog friendly. N of Valmont is no dogs.
Those are all dirt trails. Once you get into paved bike paths, there are a whole other set of options.
Finally, the roads North of the Boulder Reservoir, where the Boulder Backroads mary and half-mary are held, are very popular with runners since they are mostly flat, dirt and have minimal traffic. But, they have cars, so maybe not great for a dog. You can run most of the trail around the Reservoir with a dog, but there is a section that is no dogs…so you need to either go back the way you came, or take your chances with getting caught (unlikely during off-hours, very likely during official hours).