I play Femme and target a Data Raven.

I pay one when I hit the Raven and?:

-NOTHING on the Data Raven card happens at all

or

-I have to take a tag or end the run, but the subroutine does not execute.

It seems to read [pay X, X=Number of Subroutines on the ice] and [bypass the ice card completely]. So, if you play it on Archer, you pay nothing and go at it normally OR play four and ignore Archer's existance for the run.

You're correct on how it works regarding bypass costs (1/subroutine).

With regards to Tollbooth or Data Raven:

Femme's bypass and Data Raven both share the same trigger "when you encounter". This means runner abilities resolve first, as the runner is the active player. If you choose to bypass, then any remaining unresolved "when you encounter" abilities do not fire (see the FAQ for confirmation).

Thank You

But you did still encounter it, even though you bypassed it, so it switches off things that specify "the first ice encountered" like Kit's Code Gate ability and Inside Job. (Imagine if someone ran with Inside Job on a server with 2 ice and a Femme token on the outermost; they cannot bypass it and enjoy the benefit of Inside Job on the next piece of ice.)

But you did still encounter it, even though you bypassed it, so it switches off things that specify "the first ice encountered" like Kit's Code Gate ability and Inside Job. (Imagine if someone ran with Inside Job on a server with 2 ice and a Femme token on the outermost; they cannot bypass it and enjoy the benefit of Inside Job on the next piece of ice.)

very true. Also, worth noting, is that if you choose a piece of ICE and then your opponent plays, say, a Chum in front of it. If you let the Chum resolve, and then use Femme to bypass the following ICE, you still take damage from Chum as you didn't break the subroutines on the ICE after encountering it.