Very new to raising chickens.....Need help!!!!

We have been wanting chickens for a long time. Last week we received 8 hens that are around 26-27 wks old. They needed to rehome them asap because of there situation, so they were given to us. They were all in one 4x4 box there whole life, on top of each other and most had missing back feathers. (They had never laid an egg either) We took them , the same day we built a 2 story coop with a nesting box for each of them. We built a water system with the nipples and hung a huge feeder. I am feeding them a really good layer feed 24/7 and scratch with grit and oyster shell 2xs a day. They seem to be very happy! Due to our weather and them being a surprise 8 days ago we haven't been able to put a run up, but we will be building there run this Friday. I have done a ton of research so I can do this right, but still have a lot to learn. Yesterday morning I got my first egg, which was so exciting!!!! I went out 3 more times yesterday before bed , and again this morning and there was nothing. When we got them, they did not no the breed so we took them to our local feed store and found out they are Easter eggers and Americana's. Today I noticed brown watery stool in the coop....They are acting normal and all came up for there loving like normal but should I be concerned? Will they all follow suit and start laying? Also when I got my first egg yesterday it wasn't in a nesting box it was on the floor in the corner of the coop, is this normal for a first egg? I also have a few that peck the others bad, so have ordered pin less peepers to put on them is this a good idea? I know it is very important to let them get out of the coop and get fresh air, but with out a run yet I am very concerned with letting them roam. We live in the eastern plains of Colorado and on a lot of property, other ranchers that live near us and also have chickens have warned us not to let them free range because a lot of them have lost chickens and other animals out here because of the over population of predators. With them not having a run yet, will that hinder egg production? All I want is my girls to be healthy, happy and great layers! I want to make sure I am doing everything I can to make sure that happens. Any advise or knowledge would be great! Thank you in advance!

How big is the space they're in now? They might feel a little crowded still. How big is the run going to be?

It's pretty normal for new layers to put an egg somewhere that looks nice, but isn't a nesting box. There are various things you can do. Block off the corner by making it more rounded, put some fake eggs in the nesting box, make the nesting box more like the place they like to lay eggs...

Do you have any photos of the offending stool? Failing that, you could google 'watery chicken droppings' or whatever and point us at the closest result.

A change in diet can cause some digestive upset. The picking is from being crowded so get them more room, as much as possible very soon, you can make a smaller secure run attached to a larger less secure so they can range more when you can watch them.

They are still getting used to their new home and they won't have a clue where to lay an egg, they often don't know what is happening initially, and will drop eggs. So the fake eggs will let them know where to lay.

Easter eggers can good or poor layers based on the breeding behind them.

I've never tried the peepers, I find most stop picking after given adequate room and things to peck and scratch at.

You've gotten some good advice and questions already.

About every 8-10 poops they release a cecal poop...it looks, and smells, much different from their normal poop.
Here's some examples of the different kinds of normal poops....don't jump to any 'disease'conclusions too quickly.
http://www.the-chicken-chick.com/2012/02/whats-scoop-on-chicken-poop-digestive.html
http://chat.allotment-garden.org/?topic=17568.0

Don't mix the scratch, grit, and oyster shell.
Give them just a bit of scratch everyday.....tho it does dilute the usually 16% protein of layer feed.
Grit and oyster shell should be always available in a separate containers to use as they need, which won't be often.
The layer feed should be available all day as well.