There are some special cases where the CSM classes of singular varieties can be computed easily.
For example there is Ehler's formula for $c_{SM}(X)$ where $X$ is any complete toric variety. Let $\Sigma$ be the fan of $X$ with torus orbits $B_\sigma$ for $\sigma \in \Sigma$, then the CSM class of $X$ is given by
$$
c_{SM}(X) = \sum_{\sigma \in \Sigma}[\overline{B}_\sigma] \in A_*(X)
$$
where $\overline{B}_\sigma$ are closures of the torus orbits.
Paolo Aluffi has done a lot of work on computing these things. For example (see here) he has constructed a generalization that lives in proChow groups instead of the usual chow groups. The proChow group of a variety $X$ agrees with the Chow group when $X$ is proper and the proCSM class of a not necessarily complete toric variety still satisfies the same formula as above. This also generalizes to other varieties with stratification by smooth locally closed subvarieties but the formula gets more complicated.
In another direction, if we have an embedding $i : X \subset M$ where $M$ is smooth, then we can compute $i_*c_{SM}(X)$ in $M$ by the following formula.
$$
i_*c_{SM}(X) = c(TM) \cap [M] - \pi_*\left(c(\Omega^1_{\tilde{M}}(log X')^\vee)\cap [\widetilde{M}]\right) \in A_*(M)
$$
where $\pi : \widetilde{M} \to M$ is an isomorphism away from $X$ and $X' = \pi^{-1}(X)_{red}$ is an snc divisor. See this paper.
When $X$ is itself a hypersurface in $M$ there is a much more computable formula:
$$
c_{SM}(X) = c_F(X) + c(TM) \cap \left(\frac{1}{c(\mathcal{O}(X))}\cdot (s(Y,M)^\vee \otimes \mathcal{O}(X))\right)
$$
where $c_F$ is the Chern-Fulton class and $s$ is the Segre class and $Y$ is the singular locus of $X$. Each of the pieces of this formula are in general much more computable then $c_{SM}$ itself in the sense that there are algorithms and Macaulay2 codes for doing this. For more on this aspect and especially techniques and examples of these computations, check out this survey.
As for your second question, it is true in general that the degree of the zero dimensional piece of $c_{SM}(X)$ is the Euler characteristic,
$$
\int c_{SM}(X) = \chi(X).
$$
So the answer to your second question is yes.