Outdoor seating ok with dogs
Panevino on route 10 in Livingston. Has outdoor patio that is dog friendly. Call for reservations and mention you are bringing a dog. I was just there this past weekend and noticed they had a dog friendly sign up for their outdoor area (has plenty of umbrellas for shade)
My dog and I used to spend a lot of time in the courtyard behind Charlie Brown's in Millburn. Huge trees and almost completely enclosed, so no traffic noise or fumes. I don't think anybody knew about it; there were hardly ever any customers sitting there. I would get a slice from La Strada and Charlie Brown waitstaff would bring my dog steak scraps. Charlie Brown's is closed now, but you might check with EVOO and Lemon, which opens onto the same courtyard and I think has its own patio.
I'm pretty sure the tables and chairs in the courtyard are provided by the town for anyone to use. If so, you could get take-out from the Vietnamese place or any of the adjacent restaurants and have a nice picnic.
I have taken my dog to the La Strada patio, but not at busy times. They said yes but seemed lukewarm.
My dog and I also enjoyed the patio at Stony's. Good umbrellas and good accessibility. I haven't been there since it changed to Miti Miti, so not sure if they are dog-friendly now.
Most restaurants that have sidewalk seating (Bunny's, the SO diner, Falafel-ly Yours in SO) will allow dogs, but it's not that pleasant since you're sitting right by the street and you're roasting if it's hot, even if they have umbrellas.
As far as I'm concerned, M/SO has a dire shortage of dog-friendly restaurants. 
Need more beer gardens.
Hey, I wonder if McLoone's allows dogs on their patio. You might give them a call.
LUNCH: Family visiting with dog. Dog must come with [wonderful young black lab] Restaurant must also be handicapped accessible with shade. Suggestions?
TIA!