The Baireness of $C_k(X)=C_k(X,\mathbb{R})$ has been characterized for some subclasses of $k$-spaces. For example, Gruenhage and Ma proved that for a locally compact or first-countable space $X$, the following are equivalent:

1. $C_k(X)$ is Baire.
2. $X$ has the Moving Off Property.

Moving Off Property means that if $\mathcal{K}$ is a collection of compact subsets of $X$ such that every compact subset of $X$ is disjoint from some member of $\mathcal{K}$ (such a collection is called "Moving Off"), then $\mathcal{K}$ contains an infinite subcollection with a discrete open expansion.

(1) --> (2) is easily seen to be true in general (hint: if $\mathcal{K}$ is a moving off collection then $O_n=\{f \in C_k(X): (\exists K \in \mathcal{K}) (f(K)>n)\}$ is a dense open subset of $C_k(X))$), but it is an open problem whether the above characterization holds for every (completely regular) space $X$.

Regarding Cech-completeness of $C_k(X)$, McCoy proved that for a first-countable space $X$, the following are equivalent:

1. $C_k(X)$ is Cech-complete.
2. $C_k(X)$ is completely metrizable.
3. $X$ is hemicompact.

In the last section of their paper, Gruenhage and Ma offer an example of a locally compact first-countable space $X$ such that $C_k(X)$ is Baire but not weakly $\alpha$-favorable. Since every completely metrizable space is weakly $\alpha$-favorable, it is clear from McCoy's characterization that $C_k(X)$ cannot be Cech-complete in this case, and this solves your Problem 1.

The Baireness of $C_k(X)=C_k(X,\mathbb{R})$ has been characterized for some subclasses of $k$-spaces. For example, Gruenhage and Ma proved that for a locally compact or first-countable space $X$, the following are equivalent:

1. $C_k(X)$ is Baire.
2. $X$ has the Moving Off Property.

Moving Off Property means that if $\mathcal{K}$ is a collection of compact subsets of $X$ such that every compact subset of $X$ is disjoint from some member of $\mathcal{K}$ (such a collection is called "Moving Off"), then $\mathcal{K}$ contains an infinite subcollection with a discrete open expansion.

(1) --> (2) is easily seen to be true, even without the assumption that $X$ is locally compact or first-countable (hint: if $\mathcal{K}$ is a moving off collection then $O_n=\{f \in C_k(X): (\exists K \in \mathcal{K}) (f(K)>n)\}$ is a dense open subset of $C_k(X))$), but it is an open problem whether the above characterization holds for every (completely regular) space $X$.

In the last section of their paper, Gruenhage and Ma offer an example of a locally compact space $X$ such that $C_k(X)$ is Baire but not weakly $\alpha$-favorable. Since every Cech-complete space is weakly $\alpha$-favorable, their example solves your Problem 1.