Notation:$\sigma_e(A)$ is the essential spectrum of $A$.
In the proof of Lemma 2.1.1(i), the author mentions that $\chi_{\lambda}(A)$ can be decomposed into a sum of mutually orthogonal infinite trace projections?
How to construct the mutually orthogonal infinite trace projections?
The author also mention a fact: every projection is a sum of finite trace projections. I cannot find the proof in any reference books. Is the fact trivial?
All you need to see is that any projection majorizes a projection with finite trace. This is because of semifinitness. Indeed, given a projection $p$ by semifinitness there exists $x\geq0$ with $x\leq p$ and $\tau(x)<\infty$ (see here for equivalent formulations of semifinitness). By functional calculus there exists $\delta>0$ and $q$ such that $\delta q\leq x\leq p$. This forces $q\leq p$, and $\tau(q)\leq \delta^{-1}\tau(x)<\infty$.
Now you iterate this. If $\tau(p)=\infty$, there exists $p_1\leq p$ with $\tau(p_1)<\infty$. Then $\tau(p-p_1)=\infty$, and the argument can be repeated. We get a pairwise orthogonal family of finite-trace projections below $p$. We can then use Zorn's Lemma to get a maximal family, and then $p=\sum_jp_j$ with $\tau(p_j)<\infty$ for all $j$.
Finally, divide the infinite index set of the $j$ in $N$ sets $J_1,\ldots,J_N$, and put $$ Q_k=\sum_{j\in J_k}p_j. $$