I think you are asking about the methods to get to the summit - auto or cog. Both take you to the same place; but the only way to do either one-way would be to pay for a car service to drive one-way on the auto road. The train tickets are roundtrip. If you’re asking about the actual track; it’s the same route up and down. Hope this helps.

While there is parking below, it is considerably cheaper to parallel park (metered) on Mt. Washington. I tried parking in the lot below and would still have needed to walk a good distance to the incline.

There is a tram that goes up the side of Mount Washington from downtown that's pretty cool to do, then once up there you can walk around and get a great view of the city, have something to eat, then return. Check the schedule so you aren't stranded up there and have to take a cab back though. Or figure out the bus system (no idea on that) or take a cab/uber. Night is truly beautiful from up there - make sure to visit that chapel that's lit up at night too.

Yes. There was a stroller in the car when we rode it.

A dog rode up the tram with me, so I guess the answer would be yes.

On game days the Gateway fleet runs from Heinz Field to Station Square. From Station Square take the Incline up to Mt. Washington.

The plan seems fine. Wanted to add few more points to your trip. Try to park near Monongahela Incline as parking near Duquesne incline would be bit tight (not much of parking space). Try to walk down few yards down the hill to see the "grand view" (view point). Taking sub to point state park is the best decision considering your trip on a weekday, else you can try park at point state park as well. Check the subway route for station square to point state park (I am not very verse with Pitt subway). If you are hitting stations Square you can also try "Ducky tour" Or "Molly Trolley" Or "Evening River Cruz". If you have some time in the evening you can try drive down to north shore and walk down to the PNC park area and city streets .... This will be a charming experience in a summer evening ....

Across the street

Also consider the East End Overlook or Grandview Park

Cute home town ones and nice fine dining ones, too. Best to get a restaurant on Grandview