Translation:My head hurts because I have a cold.
You are right. In a string of verbs connected by "...te" forms, the only things strictly expressed are that the verbs are in chronological order and that their aspect/tense is taken from the final verb in the series. The entire series may be coordinate (.... and. . and.. and....) or the earlier verbs may have some logically subordinate relationship to following verbs (after, because, by, etc.). Adversative or contrastive relationships (although) have to be expressed specifically.
It reminds me of a lovely lady on youtube explaining how she heard non-native speaker ask ここに座ってもいいですか。which was perfect Japanese but a native speaker would merely say ここはいいですか。
The point is you can learn perfectly precise Japanese with every nuance stated explicitly, or you can just learn to speak the way they actually do in Japan.
The suffix "-te" is a conjunctive form (renyookei) used in modern spoken Japanese to link verbs in series. 1. The verbs in series are in order of occurrence. (First happens first, second happens second, etc.) 2. The aspect/tense of the whole series depends on the final verb to which the series is connected.
The final verb is the head of the structure. Verbs at the beginning of the series are syntactically subordinate to the verbs that follow them.
Logically, "te" form verbs in series can be coordinate or subordinate. That is, it is hard to say that it is wrong to translate "te" by "and" as a general rule---- when that makes sense because the verbs simply list a series of events. Sometimes, though, the earlier verbs are logically subordinate to following verbs. That is, they may be the reason, or the manner, or the means to the action in the main verb. There is nothing in the "te" form itself to indicate this logic. It has to be drawn the sense of the sentence.
Thus, "kaze wo hiite, atama ga itai" is literally "having caught cold, head hurts" and it is obvious that the cold is the reason for the hurting head. "I have a cold and my head hurts" is not wrong. "My head hurts because I have a cold" simply makes the logic explicit. All of this doesn't make much difference in this case but in extended discourse ".....and...... and.... and... " can make for really unreadable English.
I actually don't agree that the wording provided by Duolingo is the best possible, but it is one of the ways to translate the sentence. It's important to note that the linking form can be used this way for cause and effect, but I feel like if you use 'because,' it becomes too definite for learning the versatility. Weak causation such as, "I caught a cold, so my head hurts," might be a bit more accurate. If I were asked to translate the answer sentence to Japanese, it would come out like「かぜをひいたので、あたまがいたいです。」or「あたまがいたいのはかぜをひいたからです。」
Except it is used as "because" so it's important to learn to recognise it. But just for the record, te can be used for: sequential events (ate my breakfast, went to school); parallel events (i ate breakfast, my sister stayed in bed) and a cause and effect as in this exercise. Also te by itself is a casual imperative (mite! -- look!)
I would think more technically correct is I have some cold head pain i.e. head pain associated with a cold which is more forceful joined than the given translation. Note not necessarily a true headache rather dehydration head pain or excessive coughing shaking your head.