Well, not exactly culture, but here is some information that might make your anime-watching more enjoyable and satisfying.
In Akai Hayate, [one of the guys] is referred to as Tengu. A tengu is a Japanese mythical goblin who was a protector of the forest of sorts. They were characterized by a very tall figure, the skin was either red or blue, and the most prominent feature was their big nose (that was why [his shadow suit?] had a beak-like object sticking out).
Why is the series called Akai Hayate anyway? Well, female ninja were termed "red," and the Japanese word for red is "akai." Thus, Hayate in the form of a woman = his sister.
Have you wondered where Jinnai got the names for his Bugrom minions? Namihei, Masuo, etc., are all characters of the longest-running (more than 30 years!) TV anime series in Japan, called Sazae-san. It's also the most-watched series. It's a family-type anime - after all, it's shown at dinner time - and it's about the daily activities of a family and their friends. Think of it as another version of For Better or for Worse. Except all their names are derived from fish or other ocean terms. For example, Ikura is salmon roe.
If you might notice other things or just have a general question, drop it in the suggestion box!