Wow, you've got to feel sorry for some people. Fancy getting so worked up over grass that you wrote that note?
Weeds and chcikens go hand in hand. I'm on 2.5 acres with lots of garden so lots of weeds. Feeding the chickens gives me a great incentive to get out every day and pick some of them.
I have tractors without a door on the night house and the easiest thing it to attach a wire floor. I have two with the wire being the grade of concrete reinforcing so the chickens can still scratch through it and help themselves to the green they are parked on. I have one with a finer grade wire...
That coop might hold 12 bantams but that doesn't mean it will comfortably house 12 bantams. The more crowded it is the more conflict you will have between your chickens. I wouldn't put more than 3 standard fowl in there.
I'm just adding my voice to those commenting on the hassles of using a wire floor. I have a wire floor in one of my tractors and the poop just clogs it up like mad. The only way to get it clean is to turn the tractor over and hose it. I've got sugar cane mulch over it to help the clean up but...
I also have a wire grid under my tractors to stop predators digging in under it and it would flatten long grass and make it hard for the chickens to get at. They do have a really good dig away aroound the grass looking for bugs. I don't know if different grasses recover from that differently...
I mow my grass despite the tractors but I am on 2.5 acres and have three average sized tractors. I move them daily and mostly within a fairly small area but the chickens would still have problems keeping the grass short. It looks rotten where the tractors have been for 3 or 4 days then it looks...
I've used both sorts and they can both leak. I had no trouble getting the chicks to use them. I just touched a chick or two's beak to the ball bearing so they got some water and they figured it out from there. I had one batch of chicks who worked it out without me even doing that.
I've seen some people advocate leaving the feeder empty one day in four to encourage the grils to clean up after themselves. You could try that. Oh and I don't think feeding ground shells encourages egg eating. Mine will egg eat if they run out of shell grit but not otherwise.
A movable run is a great idea but you'll need to plan for a couple of things. They'll need somewhere to lay and tend to be creatures of habit when it comes to laying. They are usually done by early afternoon. Also they'll need shelter in case of sudden change in weather or heat. And they'll need...
I'll be saying "Sorry, my eggs are great. Feel free to find some not fed corn elsewhere." It isn't like feeding them corn is loading them up with hormones or chemicals or some other sort of questionable practise.
It depends on what you want to acheive. Some people will leave it in one place for ages to allow the poop to accumulate then let it age and use it as a vegetable patch. I move mine daily so the chickens get fresh greens each day and the ground they have been on gets to recover. I am on 2.5 acres...
My whole run is secure so I don't shut the coop at night and it works fine. You may want to look at your run security as kari_dawn suggested. I would be putting a wire apron on the ground around it and anchoring it down so no one can dig under. It is easy (no digging) and works.
So long as it is lower than the roosts it should be fine. Your girls will be able to get into it easily and shouldn't take to sleeping in it at that height.
I use a commercially available seed mix, which is Australian so you won't be ble to buy it but it has a good mix of stuff in it. It goes by the great name of "clucker tucker".
http://www.greenharvest.com.au/seeds/poultry_forage.html
A lot of people won't mix babies with big girls until they are fully grown. Last time I did it i put them in a 22 weeks but I guess you are in a bit of a bind if your old girl is by herself. One thing I have heard people do is fence of an area with gaps big enough for the littlies to go through...