Some of the instructions for wire nuts claim it is not necessary to twist the wires prior to installing the nut. My father, who was a residential and industrial electrician for 40+ years, always twisted the wires prior to installing the nut. In all that time, he never had a connection failure. The connection could always be inspected to make sure a wire didn't slide back, etc. Use a good quality nut. (you get what you pay for) Properly done, there is no need for electrical tape over the nut. The nut should feel solid and tight after it is installed.
I would check with your local inspecror as to the requirement for the boxes be exposed in the attic.