umm I dont think they need oats :-DDD just hay and a mineral salt lick (rockies do a yellow one, you dont want the tubs with molasses in).
Once they get lami they will be lame, yes but early warning signs are overweight pony, esp if 'cresty' on the neck or fat pads on the shoulders, any heat in the hooves, a racing pulse in the hoof (farrier can show you how to check for this - they will need their feet trimming if not on a rocky mountain if they arent being ridden, tho they wont need shoes).
If they still gain weight or are too fat with the hay (you might not be able to give them unlimited hay even if they get too tubby), then oat straw is a good option as it is lower in sugars/energy than unsoaked hay. The grazing while it wont have much goodness in it at the mo, if big in quantity will make them overweight, which while it doesnt directly cause lami, does make the other triggers a lot more likely to cause it. Very like type 2 diabetes in humans.
Yes, hooves should be picked out every day, and a quick check over, altho they may not need brushing everyday. If you are brushing them, try to limit how often and how vigourously you do it as if they are out unrugged they need the oils in their coat for waterproofing and very frequent in depth brushing can remove this.