"Walking" assumes you're walking. In the background, I believe it's simply restricting streets based on an attribute in the street feature class that indicates that pedestrians are forbidden. I agree that the naming convention is a bit confusing. For a pedestrian analysis, "Walking" should be checked.
To eliminate one-ways, just turn the one-way restriction off (assuming there is one). If one-way is unchecked, one-way values will be ignored.
If you solve your analysis using a distance measure (such as Miles or Meters), the solution will be optimized based on network distance rather than travel time. The travel time impedance attributes are generally designed for cars, and they use the traffic data and speed limits. However, the distance-based impedance attributes are ignoring all that and only using distance in the optimization, so it can be used for a pedestrian analysis.
The main limitation you will run into with your network will be pedestrian paths or cut-throughs that aren't included. Pedestrians can walk on paths, across parks, through buildings, etc. However, the basic street network with a distance impedance attribute will give you something reasonable to work with. You could certainly make your own network dataset designed for pedestrian analysis if you have the data.