Given these conditions... $P(x) = x$ is a cow, $Q(x) = x$ makes milk, $R(x,y) =$ both $x$ and $y$ are the same object.
This expression says the following.. $$(\exists x)[P(x) \wedge Q(x)]$$ and another $$(\exists x)[P(x)] \wedge (\exists x)[Q(x)]$$
I translated those to the following... 1. Some cows produce milk. 2. There are some cows that produce milk.
Are these correct?
On
I would answer it as follows:
$$(\exists x)[P(x)\land Q(x)]$$
means, in the context of your statements, that (note this is a very strict interpretation) there exists a cow that makes milk (maybe not more than one but perhaps).
Now,
$$(\exists)[P(x)]\land (\exists)[Q(x)]$$
means that there exists a cow and there exists a milk maker.
The first is basically correct, though I would prefer "A cow makes milk." Your formulation implies that there are multiple cows, i.e. at least two cows, that make milk, while the logic statement says at least one cow. So I prefer my formulation. Perhaps "There is at least one cow that makes milk" is the most clear, though somewhat clumsy.
For (2): There is a cow and something makes milk.