Sod will need way too much sun. You are likely to need to go with finer fescue like creeping red or others. May want to listen to this. https://ext.vt.edu/lawn-garden/turfandgardentips/tips/turfgrass-varieties-mid-atlantic-lawns-2008.html#
Also, some peat and dried and treated manure can do great for fertilizer alternatives. Corn gluten meal can deter weeds once you have a good lawn(not before) and I recommend compost as well. Aerate in the fall, and overseed again then. Fall is the best time to plant! One potential challenge will be fungus and I do not know how to keep that away other than chemicals which I would steer clear of with dogs. Even natural treatments for fungus can be a problem. Weeding is something I would do by hand or just work with after the lawn is doing well. Because it is shaded you may want to test the ph level as well.
if you do go with thicker fescue, mow as high as you are comfortable with, but at least 3 inches . For bugs I like permethrin, but use carefully