Do cows produce milk excessively?
The average domesticated dairy cow produces far more milk than would be required to feed their calf. All cows, wild and domesticated, will only lactate in the period between their calf's birth and weaning. Milk is calf-food, and when there's no calf, there's no evolutionary advantage in producing milk. 

On dairy farms, cows are milked twice daily, from spring (when they give birth) until late autumn. This mechanical milking 'fools' the cow into continuing to lactate. When a cow has stopped lactating, they will only start again after giving birth. This means that you can't just start milking a cow and expect to get milk.

Generally farmers milk their cows from spring (birth) until late autumn. The reason for stopping in autumn is simply because the grass grows much slower in winter, so there isn't enough food to support lactating cows. However, it is entirely possible to milk longer than a year; I know of farmers who milk their cows continuously for two years. These farmers will have to purchase a lot of supplimental food (like hay or silage) during winter. The advantage of milking for longer than one season is that the cows do not have to give birth every spring, but instead only every second spring.

I believe (but can't guarantee) that in winter, most milk purchased in a shop comes from the opposite hemisphere. I do know that here in New Zealand, we export a lot of milk to northern-hemisphere countries.

If you were to suddenly stop milking a cow, they might get sick but generally they will survive. It's still something to avoid!
Although if the calf is left with their mother then the cow would be relieved.

I do not have an 'official' source for these facts. However, I grew up on a dairy farm, so this was my life.