Spearman correlation and Pearson correlation If $X$ and $Y$ has the Spearman correlation $r_s=1$, is it possible that the Pearson correlation between $X$ and $Y$ is 0?

Yes if by $1$ and $0$ you mean $\approx 1$ and $\approx 0$. Here is one example in R with a stupid outlier
x=c(1:1000)
y=c(1:999,-100000)

> cor(x,y,method="spearman")
[1] 0.994006
> cor(x,y,method="pearson")
[1] 0.03571434

