They changed the rules for equipping items in the FaQ:
"Heroes now Equip Items at the Start of their Activation
Heroes now Equip Items at the start of their Activation each turn, rather than at the start of the entire turn. This small tweak has been added to make it easier to remember and more convenient for the individual Heroes to adjust their Items each turn. Just remember that the Hero may not change the Items they have Equipped until the start of their following Activation. When in the Caverns of Cynder, the Hero must take Hits from Clothing Items after Equipping/un-Equipping Items."
Thank you for clarifications.
I suppose this is a answer to my second question: if heroes can Equip Items only at the Start of their Activation they cant change equipped items when receive an item from another hero.
But I still have doubt about when it is possible trade objects.