I decided to look up the answer. I was taught how to count in Japanese in a Judo class and we used "shi" and "shichi", as well. The short answer is that "shi" and "shichi" are use for counting and "yon" and "nana" and used when you want to say how much of something you have, e.g. four chickens or seven fish.
The way she says "nana" here is funny to me, because in my native language, "na" (না) means "no" and the way she says "nana" is the same intonation that someone would use in my language to say "no" while denying something vehemently, like, "did you eat the last cookie?" - "na na!" So whenever I have to deny something from now on, I'll say, "seven!" and confuse the hell out of people
In the manga Nana, they make a joke about how both main characters are named Nana, which means seven. Eventually, they start calling one Hachi, which means eight! She doesn't seem to like it all that much but goes along with it among friends that know them both. Great manga, by the way!
Japanese appears to have two number systems. One system which only goes from one to ten is best seen as a set of prefixes: hito, futa, mi, yo, itsu, mu, tsu, ya, kokono and to. They are frequently used with a -tsu ending, but in practical use the ending depends on the type of object being referred to. The other system is ichi, ni, san, shi, go, roku, shiji, hachi, ku and ju. This is the base for numbers of any magnitude.
japanese has a pitch accent, so it does matter, this means that some syllables are high while others are low
i'm not sure if the intonation on here is correct but watch anime and pay attention to pitch to learn it (actually anime isn't that good for learning japanese but their pitch is still correct)
also this depends on the word, not the character
As stated above, "shichi" for 7 and "shi" for 4 are usually avoided because they are a homophones for 死 shi - death.
Nana is also often used over shichi because it sounds very similar to "ichi" - one, which can get very confusing.
"shichi" is still used though when counting certain things.
It's not wrong to use shichi, just less common. And the hiragana lesson here is specifically trying to teach you the kana な na