I am used to say "Bylo nám zima". Is this wrong? Or colloquial, or some regional variation?
Why is 'we were feeling cold' not accepted in this execise while in a previous one wih similar constructuion 'bylo mi spatne' only i was feeling ill is accepted and not I was ill.
I was ill is clearly Byl jsem nemocný. Bylo mi špatně is mainly I was sick (as in eaten something bad (stomach ache, vomitting) or motion sickness, or headache or dizziness, not the state of having a disease). It can also mean that you were feeling bad during your illness. So that's why "feeling ill" was accepted and "ill" not.
As to why "feeling cold" is not accepted - I am not sure, but I thing it is just missing so I will add it.
Also note that your report contains You instead of We.