This is beautiful. I am utterly in love with your Minerva. She's a delight throughout, but it is little gestures such as these: Minerva just laughs and taps the toe of Augusta's shoe with hers that make her real and human and wonderful. Her sense of humour in particular is fabulous - we do get to see glimpses of it in canon and you really fleshed out this aspect of her personality.
The other characters are great, too, especially Augusta with her streak of self-righteousness and her deep love for Frank, Alice and Neville who are safe at home. Ouch.
It's always a joy to see Severus being competent and smart. And I do like Augusta's level-headed assessment of him: ...Severus says—slowly. The young man seems to have a flair for dramatic effect that makes Augusta want a good, stiff drink. Indeed!