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.
It's a noun. Adjectives normally end in -ý, -á, -é, -í... (except the rare short forms like "rád").
Zima (coldness), chladno (coolness, chill), teplo (warmth), horko (heat) - we could use any of these nouns here. (The translations in parentheses are literal - "Je mi zima" is literally "Coldness is to me")