For the abilities underneath the "+N to combat check", for instance, am I allowed to use them when I don't have them equipped? I believe the answer is yes but I was hoping to find documentation. I think most of these likely involve exhausting the card, which further makes me think it likely is allowed.
Page 6 of the Arkham Horror FAQ here says
Q: Do I have to be using the Bullwhip or Cross in combat to gain their special abilities?
A: No, in the case of these two cards, you do not have to use them in combat to make use of their secondary abilities.
This implies to me that any other such weapon that has a secondary ability (i.e. one that isn't +N to combat checks or something similar) can also be used similarly without being equipped.