The last three examples sound fine to me, and I think you're right that the construction is the same as in your example from Gerald Durrell's charming book. Unfortunately his sentence still sounds really odd to this American ear.I've found this expression here.
The family had all gone into town, with the exception of Leslie, who was recovering from a very severe attack of dysentery. It was his first day's convalescence and he lay on the sofa in the drawing-room as weak as a kitten, sipping iced tea and reading a large manual on ballistics.
("Birds, Beasts, and Relatives" by Gerald Durrell, Chapter 5. Cuttlefish and Crabs)
I've found some similar (in my humble opinion) examples on COCA:
"At Union headquarters, Grant had a different view of the first day's fighting."
"Soon the fishhookshaped defensive line was prepared for the next phase of the battle. That ended the first day's battle at Gettysburg."
"He was a big eight-pointer, and I had him in the canoe and back to camp before my companions came in at the end of that first day's hunt."
Does this construction work in these sentences?
"A big dog's toy" is ambiguous only under certain circumstances. "The first day's contest" isn't ambiguous because we would write or say "the first contest of the day" as "the day's first contest." I'm not sure that there is any "rule" or list of nouns for this.I still can't understand one simple thing - why the construction "a big dog's toy" can be understood in two ways and the "the first day's contest" (which seems to me pretty similar to the former) cannot?
(I understand that for the sake of unambiguity it is prudent to paraphrase the latter construction as PaulQ did, but, wouldn't it be reasonable as well to say "a big dog's toy" meaning [only] "a toy of a big dog" and "a dog's big toy" meaning "a big toy of a dog"?)
As you have rightly mentioned, the construction "first day's ~" is used (my bad )I still can't understand one simple thing - why the construction "a big dog's toy" can be understood in two ways and the "the first day's contest" (which seems to me pretty similar to the former) cannot?
Our small rule has hitherto worked perfectly. But what would happen if I tried to apply it to the "the first day(A) of a contest(B)"? Following the rule I would come out with this:I once was told to avoid giving inanimate objects possessives if possible