Why is v mléku but not v mléce
Both are correct, I think.
The word "mléčný" means "dairy", not "milk". Milk is a dairy product, but dairy also includes cheese, butter, etc.
According to this link, both are valid for locative singular (and there are also two choices for locative plural).
Why do we need to use "my milk"? In Czech version there's no "v MÉM mléku" or something like that...
You don't, "the milk" is also accepted. But it is customary to use the possessive in English here, English uses possessives much more than Czech does.