The speedbrake lever does not need to be in the ARMED position for automatic operation to occur.
Yes it does. That is the
raison d'etre for the armed position. The activation of Reverse thrust in conjunction with the conditions you describe will cause speedbrake activation irrespective of the speedbrakes being armed, but the other conditions, of themselves, will not.
For training purposes etc.
touch and go landings are normally performed with the speedbrake lever down (disarmed) so that the speedbrakes do not deploy on touchdown.