I put "Czechs do not wear shoes at home". I think it should be accepted.
It is accepted. If you encounter the same problem again, take a screenshot and file a bug report.
Why isnt it Čechů now?
Čechů is the genitive plural form, which is incorrect for the subject of the sentence.
You can use the genitive in expressions like: "Hodně Čechů nenosí doma boty." - "A lot of Czechs don't wear shoes at home." - after other expressions of quantity and after numbers five and higher. ("čtyři Češi" but "pět Čechů")
Is there a difference between "čechy" and "češi"?
The grammatical cases (see the declination in the dictionary), or Čechy can also be Bohemia.