Here are some answers:
I whipped up a half-batch last night since it was so long since I've made this (about 10 years).
Does it need to be consumed immediately upon frosting?
Does it need to be refrigerated?
Does the frosting change consistency with refrigeration?
No.
No.
No.
I grew up with this icing used on a strawberry torte...an egg base genoise, split and layered with sweetened sliced strawberries on top and bottom and then covered with this icing and ground nuts.
It sat out all day in the summer heat with no problem. If there was any left, Mom would put it in the frig , but that was for the strawberries sake, not the icing. I never noticed a difference in texture.
Running my test batch, it took ~15 minutes for the flour to thicken in the double boiler and another 3 minutes to mound on the whisk. It thickens even more as it cools.
The sugar and shortening took ~10 minutes to dissolve the sugar and remove the grittiness. You want to beat it long enough because once you add the flour mixture, you're done.
The icing is light and fluffy, but not overtly sweet. It DOES however tend to leave a slightly greasy taste in your mouth, due to the lower melting point of shortening to butter. I notice this same taste sensation with bakery icing, so they may use some form of shortening in their icing.
I think it would depend on what you were icing. If the cake is very delicate, this might overwhelm it. It held its own with the eggcake/strawberries/nut combo and with chocolate cakes.
I put it in a baggie with a pastry tip and piped out some decorations. It held its shape, but I didn't test it overnight. However, I don't think the piped edges are as crisp as a cold buttercream.
I took photos along the way and then couldn't download them. I'll check tonight and see if I can put them online to document the textures.