Cryptographic properties of field multiplication While reading about AES-GCM, I discovered there is a multiplication over $\operatorname{GF}(2^{128}$). My question is about its cryptographic properties, such as:

Take a random element $X$ from $\operatorname{GF}(2^{128}$) (which is not $0$ or $1$). Multiply it with a constant $Y$ (say, the plaintext, which is defined over the same field, $Y\neq 0$). Given the product $XY$, is it possible to recover $Y$ (or some non-trivial information about $Y$)?
If the product $XY$ satifies some regularity condition (such as $XY=1$), is it possible to gain information on $Y$?
If multiple conditions like this are given, is it possible to gain information on $Y$? I mean, say for randomly chosen $X_i$'s, the product $X_iY$'s are given.

If the above problems are hard to solve, probably finite field multiplication can be used as a method for masking countermeasure for side channel (at least theoretically).
You're asking about the multiplication operation over $GF(2^{128})$; it turns out that, if we exclude the element 0, then the multiplication operation over that modified set of $2^{128}-1$ elements is a group operation; for example, inverses exist.
And, for any group operation $\odot$, we have:

For any element $Y$, if we multiply it by a random (and independent) element $X$, the result $X \odot Y$ is random (and revealing it does not reveal anything about $Y$).

This answers your question one (with your question being modified by having $X$ exclude the 0 element only, not the 1 element)
.  Remember, 0 is not a member of the group (even though it is a member of the field; we deliberately excluded it when defining the group); in contrast, 1 is a member of the group, and avoiding it would leak some information, in particular, the value that $Y$ is not (because $Y \ne X \odot Y$
)

If both $X$ and $Y$ are unknown, then revealing $X \odot Y$ does not reveal any information about $Y$

This answers your question two.

Revealing $X_i \odot Y$ for a large number of random $X_i$ values also does not reveal any information about $Y$.

This answers your question three.

If the above problems are hard to solve, probably finite field multiplication can be used as a method for masking countermeasure for side channel (at least theoretically).

Possibly, but it wouldn't be ideal.  For one, we had to deliberately exclude the 0 value; a real implementation that does masking can't ignore such a possible value; in particular, if you had a $Y=0$ value, masking wouldn't work in that case.  On a more practical note, masking would usually involve computing inverses, and while computing multiplicative inverses over $GF(2^{128})$ isn't that hard, it isn't exceptionally trivial either.