If I phone at 8 AM on July 27 to purchase tickets for Long House and none are available, is it still possible for my car of 4 people to enter the park and walk trails, see Step House, etc?

You can enter the park and walk trails, including trails to Step House and the Long House Overlook, without advance planning. There is an entry fee for the park. The Visitors Center, Museum, and many indoor facilities are closed, although the Visitors Center book store was open.

Yes

When we were there this week, the road to Long House was closed for repairs

I was just there a few days ago, no pre purchase, did self tour and was able to drive around and walk around no problem

I don't know. My husband purchased ticket online as a surprise. Have you tried looking online?