Sorry I cannot comment since reputation is not high enough for that but my answer to your question might not be perfect.

“I had a cold last week.” Should this be translated using the imparfait “J’avais un rhume la semaine dernière” or the passé composé “J’ai eu un rhume la semaine dernière”?

As a native speaker I would use “J’avais un rhume la semaine dernière”. But in this case it seems a bit wobbly… Even though the imparfait in this situation is correct, there are better ways to say this. After asking my friend he told me he would translate it like this “J’ai attrapé froid la semaine dernière ?”.

Does the answer change if it is instead “I had a cold for three days last week”? Would it be “J’ai eu un rhume pendant trois jours la semaine dernière”?

I would use “J’ai eu un rhume pendant trois jours la semaine dernière” or even better ” J’ai eu un rhume qui a duré trois jours la semaine dernière”.

So in the end, try not to stick too much with proper translations but try to learn expressions which is what I did when I learned french. But I think that in the end you got the fact that imparfait and passé composé change depending on the time situation (since, for, ..)

“I had a cold last week” usually translates to J’ai eu un rhume la semaine dernière, i.e. last week, there was an event and that event was that I had a cold. You still use the passé composé if something else happened after the cold started: (J’ai eu un rhume la semaine dernière et puis j’ai eu une migraine) and of course if answering to the question: Qu’est ce que tu as eu la semaine dernière ?.

You might use J’avais un rhume la semaine dernière if the cold lasted the whole week (or more), or if telling something else unrelated happened during the cold (J’avais un rhume la semaine dernière quand tu m’a téléphoné) and of course if answering to the question: Qu’est ce que tu avais la semaine dernière ?

If the cold spanned for a precise period of time, that should be J’ai eu un rhume pendant trois jours la semaine dernière, not j’avais un rhume pendant trois jours la semaine dernière