Some of the example sentences have unlearned kanji omitted, yes. You’ll see the third sentence will have more kanji, and the first is usually the easiest sentence to read.
しまう is often in kana, though! You’ll hardly ever see 仕舞う in the wild.
It’s the same sentence. WK, just like a lot of simple Japanese reading material, will leave out kanji presumed to be unknown or too complex and replace it with kana. The sentence itself doesn’t change though:
自転車で転けて、前歯欠けて仕舞った.
If you wanted to you could in theory write the entire sentence in kana, you would just end up making it a lot harder to read.
I haven’t actually ever seen either 仕舞った or 転けて in kanji though.
Omitting kanji is not a matter of correct or incorrect, but rather one of style. In the most formal documents, they might put everything in kanji as you did (although they would also use different vocabulary), but in most “regular Japanese” sentences, words like しまう are basically never written in kanji.
Now the weird part in example sentences like this is where words like 自転車 are written with partial kanji - this kind of spelling can be seen in the real word, such as in 子ども - but for the most part words are either written fully in kanji or fully in kana.
They switch to writing it as 自転車 from level 10 on after they’ve thought 転. They use the same style of writing as a children’s book would use early on, so you might encounter some words in partial kanji.
You can search in kana only, it’s the second こける that comes up in a kana search it seems. So even if you don’t know the kanji for a word you can still use it to search.
Jisho does list it as “usually written using kana alone” (they list it after the definition), so it doesn’t appear to be that uncommon to write it in kana.
I guess we could write する as 為る every single time…
In all seriousness though, some words are very rarely (if ever) written in kanji, and it’s actually much more natural and daresay even correct to write them in kana instead.
To be perfectly clear, writing a word that is usually written with kanji entirely in kana is never “wrong”. In fact, there have been some contexts where Japanese used to be written with no kanji at all: e. g. the Tale of Genji was, to my knowledge, written entirely in Hiragana as it was considered inappropriate for women to learn and use kanji back in the Heian era. When computers first became capable of handling Japanese script at all, they could only handle hiragana and katakana (and quite often they had to squeeze those into the same dimension they used for fixed-space latin script, giving us half-width kana). And even in the late '90es, the first generation of Pokémon games used kana exlusively for technical reasons.
Choosing between kana and kanji is mostly convention, plus a layer of register shifting (i. e. adapting speech to the circumstances and social groups in which it is used), as has been pointed out before. Using kana is always a legitimate choice.
