I've been having trouble escaping Coldridge without killing anyone, as the achievements states 'You completed the game without killing anyone' and so I assume that means Coldridge as well. On the other hand, the Shadow achievement says, 'You completed all missions after the prologue without alerting anyone'... Is the prologue Coldridge? Can I just run through Coldridge, alerting everyone, and then not alert anyone for the rest of the game and get the achievement?

The prologue is when you are going to meet the empress. The game officially starts in the prison. So, no you can't just run through alerting everyone.

For me, it seems that this is all about luck. Also, for me, I have not once been able to get both 0 kill count and "Not Detected" in Coldridge Prison.

Even when I followed http://n4g.com/news/1098320/dishonored-achievement-walkthrough-vanished-ghost-mission-1-coldridge-prison step for step, I still got Bodies Found: 2. Other times, I get Hostiles Killed: 1 but either way, I can't get both. Not possible.

To get Clean Hands on this mission, I had to knock every guard in the prison unconscious and haul each and every one of them to the interrogation room, then I had to ghost the sewers - and down there, I had to get up on the rocks to avoid the rats (it can be done, you have to be fast) so I didn't have to feed them corpses. Dozens of other approaches and countless other attempts completely failed for me.

It's not about luck. I got both, and I wasn't even particularly trying-- but I did have to plan it out.

I found the prison runthrough really easy. As long as you get the ghost box and no kills boxes ticked you get the achievements later on I think. What I have REALLY had problems with is ghost/clean hands on the flooded district mission because the flowthough has changed on my current playthrough and now I have got to see Granny Rags for the damned sewer key so I can't ghost the damned level. Sometimes the great thing about the game is its biggest annoyance imho. Also not being able to check that summary sheet until you reach the end of the multi layer missions is a pain because you can't check where you messed up.