Are all constant fields conservative? Are all constant fields conservative? Can there be some constant vector fields which are not conservative?

The answer is affirmative. A conservative field is a vector field which is the gradient of some function. So, if $\mathbf v$ is a constant vector field, that is $\mathbf{v}(x_1,\ldots,x_n)=(a_1,\ldots,a_n)$, you can take$$F(x_1,\ldots,x_n)=a_1x_1+\cdots+a_nx_n.$$Then $\mathbf{v}=\
abla F$.