Is False Lead's effect activated when it is forfeited by another card's effect?
False Lead is not activated when forfeited by another card.  That's because the ability on False Lead is a Paid Ability, not a Conditional Ability.

A card’s trigger cost is always listed in its text box before the
effect, following the format “cost: effect.” A paid ability can be
triggered an unlimited number of times as long as the cost is
paid and any restrictions specified by the effect are observed.

By being a Paid Ability, forfeiting False Lead is an explicit cost of the ability, and must be paid in service of activating the ability.
As a result, you can't both forfeit False Lead to pay for Archer (which requires forfeiting an agenda as part of it's rez cost) and to pay for False Lead's ability.  False Lead can only be forfeited once.  Since, both cards require False Lead to be forfeited as a cost, you have to chose one.
In order for False Lead to work as you imagine, it would need to say something like: &quot;When you forfeit False Lead, the runner loses [click] [click].&quot;