Assuming you do not have a light on all night, there is no reason to feed or water them at night. Once they go to roost, they go to roost. They do need some hours of darkness also. Certain egg defects can be caused by constant light.
As long as they have access to the water shortly after they wake up, there is no reason to keep it inside. They will wake up thirsty moreso than hungry, although it's hard to find a chicken that is not hungry. If you sleep in on weekends, say, I'd have water inside. But as long as you commit to getting up early enough to let them out shortly after they wake up, the water can be outside.
I think you will find rodents whether you feed inside or outside. The rodents are going to find the food, where you store it if you don't take real strong measures to prevent that and certainly where you feed it. And snakes can always follow the rodents. I have a constant mouse trapping program ongoing here. Grown chickens love to eat mice, by the way.
Some people feed inside as they think they attract fewer wild birds that way. The more wild birds around, the more of your feed the wild birds eat plus they can carry certain parasites and diseases. Although it is not as important as water, I think it is good for them to eat soon after they wake up. So again, how much do you plan to sleep in.
As far as feeding outside, the less time they spend inside the coop the less poop load there will be in the coop for you to actively manage. Poop is usually easier to handle in the run that the coop. Most of the poop in the coop will be under the roosts so that is where you will have most of your problems and do most of your raking, but I believe in totally cleaning out the coop as seldom as I can manage. It is a lot of work in mine.
I feed layer pellets. The chickens are going to spill food. If you feed inside on litter, they cannot find all the spilled pellets so you have an inefficiency in buying them feed plus the spilled food can attract rodents inside the coop, even if you put the food away at night. My run does not have litter on it, so they can find any pellets they spill, especially since I occasionally let them run out of pellets in the feeder. I do this so they will clean up the spilled pellets and, more important to me, they will clean up all the feed in the feeder so it does not stay in there and get stale (feed loses vitamins when it gets old and stale) or mold and mildew.
Another argument to me for feeding outside is that feeders and waterers take up space. I'm not thinking about the 4 square feet per chicken type of space because if you feed and water in the run you are not leaving them locked up in the coop so the dynamics of that changes. Chickens are clumsy fliers. I like to give them a fairly open area to fly down off the roost so they won't bang into a feeder or waterer on the way down and hurt themself. I had an Australorp pullet that hurt her neck. (She recovered, by the way) She could have done it several ways, but I think a real strong possibility is that she hit a feeder I had hanging in the coop when she came off the roost. I know a couple had hit it before when I opened the pop door. It was a freak accident but I did move the feeder.
I made my pop door 12"x 12" or 3o cm x 30 cm. You can see the breeds I have in my signature. It is plenty big enough. The main reason to me for a pop door instead of leaving the human door open is to keep the elements out. In this case, smaller is better.
I'm not familiar with your plans for the coop and run. All this is predicated on your having a walk-in coop. If you have a small elevated coop or a run that has litter on the ground, then some of my reasoning would not apply to you. Take what applies to your specific situation and ignore the rest.
Good luck!!!
