I think that you're supposed to start at the top and work your way down. That way, if there's a rain storm, the water doesn't get into the wall and sit on top of your new pointing.
And don't get over aggressive when raking the old stuff out of the joints either. 1 inch into the joint is PLENTY from what I've read.
As far as the lime water staining the stone, I don't think it'll be that much an issue. But you'll want to protect the new pointing from the weather until it sets up properly. Best way to do this is to hang a canvas tarp over the area. You should spray down the canvas with water to keep the whole area damp so that the pointing doesn't dry out too quickly and crack.
As for a pro to do teh work, there's a fellow in Quakertown or Doylestown by the name of Degruchy. He is a mason that deals entirely inhistoric homes. From what I've heard he's very good. I doubt he'd agree with my recipe since he likes to use actual quick lime instead of the dry hydrated stuff, but there's two different schools of thought out there regarding that issue - besides, I can't for the life of me find the quicklime anywhere.