Sensitivity and Block sensitivity
2 does not hold as stated. All you know is that if you flip the all bits in a given block $B_i$, the output flips. Without additional conditions, I don't think you're guaranteed anything about flipping any individual bit within $B_i$. 

EDIT: In fact, if $B_1, B_2, \ldots, B_i, \ldots$ represents a maximal set of blocks, then for any $B_i$, there cannot be more than one index within $B_i$ such that flipping the bit at that index will flip the output. Suppose that within $B_i$ there are two indices, say $j$ and $k$, such that flipping either $x_j$ or $x_k$ will flip the output. Then $B_i$ can be replaced by the two singletons $j$ and $k$, thus contradicting the assumption that the set of blocks was maximal.

Meanwhile, the first function found to have a quadratic gap between $BS(f)$ and $S(f)$ is due to Rubinstein (1995). You can find a quick summary of the construction in this blog post by Scott Aaronson. A function with slightly bigger separation was found later by Virza (2011).