I don't fully understand Grassmann variables myself, but I think I can answer your question:

1. Yes, the action itself is Grassmann-valued. This is not as bad as it sounds, because in the path integral, we integrate over them, and we know how to do that with Grassmann variables. The action does not have to be real for the path integral to make sense.

In fact, bosonic fields take values in the Grassmann algebra as well. Bosonic values are always even elements of the algebra. For instance, when doing a Hubbard-Stratonovich transformation, we shift e.g. $\phi \to \phi + \overline{\psi}\psi$ where $\phi$ is a bosonic field and $\overline{\psi},\psi$ are fermionic fields. Clearly, this only makes sense when the field $\phi$ takes values in the Grassmann algebra, and is not just a real number.

That said, the action is always an even element of the Grassmann algebra, so it commutes with all other elements. In that way, it behaves more like a "number".

For a mathematically precise review of Grassmann variables and integration, I found Terence Tao's blog post "Supercommutative gaussian integration, and the gaussian unitary ensemble" extremely helpful. This question is answered there as well.

2. In a path integral setting, the classical equations of motion are obtained by looking at the stationary points of the exponent. This makes sense when the exponent is real-valued, because we can argue that the integral is dominated by the contributions from the stationary points. Unfortunately, I don't know whether this argument still makes sense for Grassmann fields! After all, we cannot argue that the integrand has a maximum, because that assumes that the integrand is a real number which can be compared with other real numbers. I don't know how to justify it.

Of course, we can still formally write down the Euler-Lagrange equations. For this, we only need to know how to perform derivatives with respect to fields. These equations now take values in the Grassmann algebra as well, and would need to be solved with this in mind. This has been done to obtain a classical variant of spin, e.g. in F.A. Berezin and S. Marinov, "Particle spin dynamics as the grassmann variant of classical mechanics" (1977). I am not sure how far this has been developed, I am unaware of any "serious" applications (and would love to be corrected on this).