As Sevenseas said, I work at a dairy farm sanctuary that meets my definition of "humane". It still can be seen as an inherently parasitic relationship but I am personally satisfied that the cows are happy and that the people who work there, and most of the people who drink their milk, are not seeing them as commodities or walking vending machines. They are sacred, revered, respected and loved. From an AR perspective there are of course still things 'wrong' with the practice and relationship but for me, my conscience is clear.
Anyway I can answer your questions with regards to our goshalla (Sanskrit for 'place where cows are protected):
...I'm just wondering if humane dairy farming is possible in order to meet demand.
No, it isn't. If ahimsa milk was made standard, people would have to drastically reduce their dairy consumption and treat milk and dairy products as precious commodities. People would have to pay the true price of milk, not the price of misery and slaughter that they currently pay for a litre of blood milk. This would be better for the cows, human health, and the whole human race IMO.
Wouldn't the cows still need to be impregnated? Would the farmer have to wait for each cow to become pregnant on her own or would they still be forcibly impregnated?
Yes, cows need to have a calf before giving milk in the vast majority of cases. It's not unheard of for a cow to spontaneously lactate, which can also happen in humans, but no farming system can rely on that freak occurrence. Artificial insemination is not sanctioned in the scriptures so we don't use it. We have a bull, Kamadeva (means Cupid) and every year he gets some new girlfriends to do it with the natural way. Sometimes he escapes and we get unexpected pregnancies, and sometimes our girls jump the fence and meet an amorous neighbouring bull. Whilst it's not their choice whether they're put with Kamadeva or not, it's certainly nothing like the rape racks you get farmers 'joking' about. I can testify that the girls really adore Kamadeva, when it's time to leave him and have their calves they really don't want to say goodbye. Unfortunately it's too risky to let them calve with him around.
The calf would drink a portion of the milk and may also be allowed to grow up. Would it be profitable enough for them to want to pursue humane dairy farming?
This is what we do. The calves and mothers stay together until the calf starts chewing the cud and doesn't rely on milk as a sole source of nutrition. Then they are put in adjacent pens so they can always see each other, touch noses, groom each other and so forth. The cow is milked and then put in with her baby. They're very clever and know to keep back enough milk to feed the baby. The calf usually needs about 25% of what she produces - we all know cows make much more milk than is necessary for their babies - and she give us the other 75%. Interestingly, cows will also hold back their milk if they don't like the milker or if the milker is nervous or angry or feeling some other unsettling emotion. I've experienced this myself, the girls wouldn't give me as much milk as they usually give until I bonded with them and got to know them.
As for profitiability, the economics are something of a problem. To account for keeping the cows in old age as well as the bulls, we need to factor in that cost in the price of the milk. At any one time about 1/3 - 1/4 of the herd is producing milk, and these girls have to pay for everybody as well as for themselves when they dry up. This is why we charge £3.50 for a litre compared to the £0.70 people pay for blood milk. A surprising number of people are fine with that, but convincing the meat-eating masses that this is what they SHOULD be paying is not easy. I spend quite a large amount of my time preaching about cow protection and really, it isn't easy.
Would they end up with too many bulls?
We have 18 oxen, 1 bull and 32 heifers/cows. For those who don't know, a heifer is a female who hasn't had a calf yet. We don't sell, slaughter or euthanise any of our cows but there are a few reasons we end up with a much higher ratio of girls: firstly, bull calves are much more likely to be stillborn, die at birth, or die in young infancy. I'm not sure why but this is common to all herds and not just our farm. Secondly, bulls tend to not live as long, much like human women have a longer life expectancy than men. When they are big enough our boys work, things like ploughing and hay-making. It stops them getting bored, exercises them, helps the ox drivers bond with them and also cuts down on our fossil fuel emissions and makes us more self-sufficient.
What would the farmers do to ensure each cow's health and avoid common problems like mastitis or milk fever? Would cows be kept alive until they die naturally even after milk production decreases with age rather than being turned into cheap ground beef?
We have two cow nurses who do a daily herd check. They will look at each cow's posture, behaviour, general condition, eyes, smell, etc. every day to catch any problems early. We have one girl, Ana, with eye cancer. She has been getting chemotherapy but it doesn't seem to be working so she might be taking a trip to Liverpool to get radiotherapy soon. The most common problems are general cuts and scrapes, which you will get in a place where their horns aren't burnt off at birth. Mastitis is a very rare problem for us because we hand milk instead of using machines, and we don't demand high yields. Our highest producer gives about 13 litres a day compared to the 20 - 30 that is common in most farms. Even our oldest girls who might have had 4 calves, the bottom of their teats never falls below the knees. Most of our cows have udders so small that you can barely see them without bending down. I have never heard of any cow having milk fever in the 30 year history of the goshalla.
None of our cows will ever go for beef, or to market, or be sold, or slaughtered, or euthanised. One cow, Haribolananda, lived until she was 28 after about 15 years of unproductive retirement. Oxen work from the ages of 4 - 15ish, the girls will generally lactate for about 6-8 years over their lifetime. The average lifespan is 18 years for our herd. We don't value them based on what they give us, they are each individual and beautiful souls who deserve the most respect and the best care we can give them. The only privileges the productive cows get is more intensively nutritious food, i.e. grains, because the demands on their body (to produce milk or work the land) are higher than the retired ones.
I imagine people would still need to limit their dairy consumption quite a bit in order for it to be sustainable and not have an overabundance of demand that they couldn't handle.
You're absolutely right, and I think that would be a very good thing for all involved.