Everyday (a 1 word phrase) means ordinary or commonplace.
Every day (a 2 word phrase) means daily.
The sentence requires the phrase ‘every day’.
In addition, the preferred word order for that sentence would be:
They ____________ to school every day when they were kids.
Assuming those corrections are made, then:
(i) is wrong. It should be ‘used to’. See here for more information.
(ii) is possible.
(iii) is possible.
(iv) is possible.
They used to cycle to school every day.
They would cycle to school every day.
They never cycled to school every day.
On reflection
‘They rarely cycled to school every day’ would not be correct.
‘They rarely cycled to school’ would be correct, but of course it cannot take ‘every day’ as a habitual phrase.
I’m not sure why I missed it first time round. Possibly because I was focussed on the incorrect original sentence.
Thank you.
They never cycled to school every day.
(Sometimes they cycled to school, but they didn’t do so every day.)
Although the usual way of voicing such an idea would be: Sometimes they cycled to school
this alternative phrasing is not incorrect.
The other sensible alternative: They didn’t cycle to school every day
is more ambiguous as you cannot tell that they cycled at all.
I have no doubt that the test writer did not consider it though. It seems to be the case that many of the multiple choice tests written by non-native teachers are inflexible with regard to the formality of the application of certain rules.
Sadly, like the original sentence in question here, there are many examples of tests which are very badly written and grammatically incorrect.