Most bottle calves are introduced to grain and hay any time after 2 weeks. In nature a calf will start to mimic momma at about 2 weeks and you will see it "grazing" which of course it isn't really eating much but it is learning what "cows do". I raise alot of calves; some on nurse cows and some bottle. I have calves that are getting into the feed trough with their momma, trying the grain by a month. And I have seen them actually eating/chewing a bit on the hay before then.
A calf should be eating 2% of it's body weight when it is weaned off milk, so that it continues to grow and not have a set back. Letting a calf dictate when it is ready to be weaned is not really reliable. I have calves that haven't wanted to be weaned at 6 months unless I push the issue. Also, too much milk and not enough concentrates and hay will cause a heifer calf to get a "fatty udder" and can actually cause her to have problems down the road.
The stomach has 4 chambers, and the roughage (hay) and grain causes the rumen to develop so that the calf can survive and thrive on grass/hay/other roughage and grain. They are designed to utilize this most efficiently. It needs to be developed early in life. Again in nature, a calf will still be nursing at 5-7 months, but they are getting very little, and they are eating grass and hay and the rumen bacteria are digesting this and turning it into fuel that the body can absorb and grow on.
Most dairy farmers are weaning their calves at 6-12 weeks; depending on their program. I like to get a calf completely off milk replacer by 12 weeks. The ones on the nurse cows will stay for about 4-5 months, but they are eating good by then. Part of it is also the expense. Good quality milk replacer is $60-75 a 50 lb bag. If you are raising very many it can be quite costly and the grain is a better investment.
Yes, you increase the grain and decrease the milk/bottle. If you are feeding alot of milk, they have no incentive to want to try much solid food either. 1/2 to 3/4 gallon milk a feeding is more than enough for most calves, and they will be chewing on anything they can find in the pen after a week or so. I cut the milk to no more than 1/2 gallon a feeding by the time they are 4-6 weeks to encourage them to try other food. Usually I will go to once a day feeding if the calf is consuming a decent amount of grain by 8 weeks; then taper off so by 12 weeks it is done with the bottle. Most dairies will allow the calf to have it free choice in the beginning, and working up to the calf getting enough and then get grain 2x a day.
The momma cow will clean the calf's butt. That said too much milk and not enough solids will cause the calf to continue to have soft manure and keep the problem going. I don't clean any butts unless they get scours and I don't want them to get scalded. If she has alot of hair I would clip it and get her on more solid feed so the manure is more solid. Occasionally, you will get a calf that is too lazy to pick up it's tail when it does manure. Again, more solid manure tends to make them naturally pick up their tail. Clip it short now, before winter, and hopefully it will not continue to be a problem as it matures.
Good luck.