All day is a fixed expression that means the same thing as the whole day. Your grammar book is correct.
All of the day is grammatical but not idiomatic in the sense of being an established expression. So your example sentence is fine. More often, people might say We spent all of that day on the beach, referring back to an occasion that they have already ...
Because you have "that" acting as a determiner, you don't need an article as well.
after that first year.
Consider as an example:
Look at the car.
Look at that car.
Both "the" and "that" in these two examples serve to highlight which particular car you mean.
"The snow" if you are talking about a particular snowfall event:
I remember the snow in October when I was 10. We built a snowman outside the school.
But if you saying that you remember "snow in October" generally, without referring to a specific event, then no article:
Nowadays the world is much warmer, but I remember snow in early ...
When you use:
For the efficient running of the business...
then "running" behaves more like a noun, being paired with an adjective ("efficient") and followed by a genitive ("of the business"). That is the main reason why the omitting of the definite article is not grammatically correct.
The definite article is needed in order ...
Your textbook is correct. Yes, we really do need the definite article there. It announces an approaching gerund: running.
Although you're right in thinking "the house" is specific and "a house" is not, there is nothing specific about the efficient running. In "the stealing of other people's property is illegal" there is no ...