Runner has a Corrorder Icebreaker that has strength 2, 1 credit break barrier subroutine, 1 credit +1 strength.
Runner runs and the corp pays to rez his first ICE.
Wall of Thrones Barrier, First subroutine Do 2 net damage, Second subroutine end the run, strength 5.
Runner pays two credits for Corrorder to break each subroutine of Wall of Thrones. And then he pays 3 credits to match Corrorder's strenth to the Wall of Thrones.
Then the runner continues to the next piece of ICE and the corp pays to rez the ICE.
Enigma Code Gate, first subroutine the runner loses click if able, second subroutine end the run, strength 2.
Runner only has Corrorder in his rig. Runner has no icebreaker to break either subroutines. Each subroutine goes off. The runner loses a click and the run ends.
The run ends and the runner had a failed run. Or, because the runner had a strength 5 Corrorder at the end of the run he was able to access the unrezzed agenda.
Another question is, I am a runner and I have an icebreaker that cannot touch an ICE type and strength. I have no way to get past the ICE but the subroutines go off and I am still alive. Do I bounce of of the ICE or do I get to pass it?
As radiskull said, you've got a few fundamental things wrong here.
You need to pump the icebreaker to the right strength before you can use it to break the ice. Same cost in your example, net result is the same, but best to get this clear.
Your example was pretty good, you explained that the corp rezzed Enigma, and the runner didnt have a code gate breaker, so he lost and click and the run ended. But then you went on to that he might still be able to access a card and that corroder was strength 5 at the end of the run Icebreakers (unless stated otherwise) only keep their increase in strength during the encounter with that ice. As soon as they have passed that ice, they revert back to normal stength.
And "end the run" means "end the run". - it ends. They do not get to access cards.
For your other question, if you cannot or do not break a subroutine, then the corp triggers it and does what it says. If it says "end the run", then the run ends. If it does not say "End the run", you can carry on.
Eg: Turn 1, Corp puts Ice in front of R&D and you run it right away with nothing in play. You take 3 net damage, but then you access a card, because the ice did not say "end the run".
Eg 2: Runner has exposed an Agenda in a server and is about to run it to win the game. Corp rezzes wall of thorns. The runner only has enough credits to break one of the two subroutines. So, he chooses to break the 'end the run' subroutine, takes 2 net damage, steals the agenda and then wins the game. i.e: You only need to break the subs which say "end the run" in order to get past the ice.