Is there a constant known to be algebraic but unknown whether it is a surd? Is there a mathematical constant known to be algebraic irrational, but which is unknown to be a surd (root of an integer polnyomial of degree $2$) or not ?
Is there a mathemetical constant known to be algebraic, but unknown to be rational or not ?

Irrationality proofs or transcendental proofs can be extremely difficult, so I wonder whether verifying a constant to be a surd or not, can be extremely difficult as well (even if it is known that it is algebraic and irrational).
I have not much hope concerning the second question because a number known to be algebraic can probably be proven relatively easy to be rational or irrational, but maybe I am wrong.
What do you mean by "mathematical constant?" Do you mean something like $\pi$ or $e$? Or do you mean something without a name like 17 or $\sqrt{2}$?

An algebraic number is a real number that is a root of a nonzero polynomial equation with integer coefficients. I would say $\phi$, the golden ratio, is a mathematical constant that is algebraic. It is irrational because it is equal to $\frac{1+\sqrt{5}}{2}$.

If I recall correctly, a surd is an nth root of a number (not just a square root, but any root), and so is a root of some polynomial. Therefore, it is irrational. So it seems to me your first question is asking, "Is there something that is an apple that is not known whether it is an apple?" Maybe I'm not really understanding what it is you're asking.

Regarding the 2nd question, we know the only numbers with rational square roots are perfect squares (in fact, the roots are integers). Similarly, the only numbers with rational cube roots are perfect cubes (again, the roots are integers). So it would seem any algebraic number (any root of a polynomial) will be known to be irrational, unless it is an integer.
What do you mean by "mathematical constant?" Do you mean something like $\pi$ or $e$? Or do you mean something without a name like 17 or $\sqrt{2}$?

An algebraic number is a real number that is a root of a nonzero polynomial equation with integer coefficients. I would say $\phi$, the golden ratio, is a mathematical constant that is algebraic. It is irrational because it is equal to $\frac{1+\sqrt{5}}{2}$.

If I recall correctly, a surd is an nth root of a number (not just a square root, but any root), and so is a root of some polynomial. Therefore, it is irrational. So it seems to me your first question is asking, "Is there something that is an apple that is not known whether it is an apple?" Maybe I'm not really understanding what it is you're asking.

Regarding the 2nd question, we know the only numbers with rational square roots are perfect squares (in fact, the roots are integers). Similarly, the only numbers with rational cube roots are perfect cubes (again, the roots are integers). So it would seem any algebraic number (any root of a polynomial) will be known to be irrational, unless it is an integer.