Answering the second question, if $K$ is shellable, then indeed 

$$|\chi (K)|\le \sum \beta_i  \le N,$$

since each shelling step either leaves all Betti numbers unchanged or else 
increases one Betti number by 1, and the number of shelling steps equals the number of facets.

Regarding the first question, here is an upper bound in terms of the number $N$ of facets and the dimension $d$ of the complex: $|\chi (K)|\le (d+1)! \cdot N$ by 

(1) Observing that the barycentric subdivision of a pure $d$-dimensional simplicial complex has $(d+1)!\cdot N$ facets if the original complex had $N$ facets (where pure means all facets have the same dimension), and removing the purity requirement only reduces the ratio in the number of facets; and 

(2) Noting that a simplicial complex $sd(K)$ having $f$ facets that is the barycentric subdivision of a simplicial complex $K$satisfies $|\chi (sd(K))| \le f$

We check (2) by using that $sd(K)$, regarded as an abstract simplicial complex, may be intepreted as the order complex of the face poset of $K$; this enables the use of a discrete Morse theory construction called ``lexicographic discrete Morse functions'' which produces for the order complex of any finite poset having unique minimal and maximal element a discrete Morse function in which each facet of the order complex contributes at most one critical cell (the discrete Morse theory analogue of a critical point, where critical cell dimension corresponds to index of a critical point).  This construction appears in a paper entitled "Discrete Morse functions from lexicographic orders".  So, the upper bound follows from the interpretation of Euler characteristic as alternating sum of number of critical cells of each dimension.