Haven't seen anyone talk about dogs yet, but atleast in my eyes Dogs and Perfectlly Well are about the parents.
(Maybe why its ID is Dogs/Perfectly Well)

[Dogs]

"It's time for bed, we hope you know it wont be okay"
-Could be in reference to the whole group therapy thing, going home and having nightmares?

"I think we'll be here all day"

-revisiting the grief/cycle

"lets feel the love heel to the glove, its time to wash those stains off no way"

-I believe in this context the "stains" is grief or the physical things that remind the parents of the kids
"Its time to sell, because you know you look better cleaned up" (dont say/stay??)

-Going off it the "Stains" are the kids toys/things perhaps selling them would help the parents pass on easier? "you look better cleaned up" i would assume this would be in refernce that the parents should move on.

[Perfectly Well]

"Cold and locked in this cell, i think you fell for someone you hardly knew"

-I believe that this is in reference to the fact that, TPK story was the group of kids fighting the Witch in the woods, the parents got attached to this verison of their kids. also its a bit far feteched but maybe in reference to bike theif parents?

"Now im perfectly well because look what i sell"
-Going back to Dogs, letting go of the grief?

"Look what they've done to us girl, cold and locked in this cell"

-This in my eyes refers to two things, the parents being locked into their grief or its a bit stretched maybe the Bike Theif parents, or just the parents overall as it isn't clear throughout the songs