While there are many ways to express the idea, the phrase ”rank them in terms of our favorite" sounds perfectly natural to me. Personally, I’d say “rank them in order of (your) preference”. We could use the past tense to describe the handout, since it was given to you in the past, but if the handout is still the same *now*, and the information is still relevant now, then it would be correct and make more sense to describe it in the present tense, even though the past tense would not be wrong. If it’s in your hand now, definitely use the present tense. Actually, we could avoid the verb “are” entirely, saying “…a handout with a list of activities…”.

Grammatically it's "were" since you're referring to something (being given the sheet) that happened in the past. A better phrasing that avoids the issue might be "gave us a sheet that listed/described many activities that we could join". Using a more descriptive verb than are/were conveys more meaning and avoids the grammatical confusion. "Rank them in terms of our favorite" sounds a little unnatural. I think I'd say "Rank your choices". While you can say that something is your "second favorite" or "third favorite", I mostly use the word for only the thing I like best.