Proving $\lceil \lg n \rceil -1 = \lfloor \lg n \rfloor$
The expression $\lceil \log_2 n \rceil - 1$ for the height of a $n$-element heap is wrong.
For $n=1$ the expression yields $-1$ instead of $0$, for $n=2$ it yields $0$ instead of $1$, etc...
In general, for $i \in \mathbb{N}^+$, an heap with $2^i$ nodes has height $i$ but $\lceil \log_2 2^i \rceil - 1 = i-1$.
The equality $\lceil \log_2n \rceil -1=\lfloor \log_2 n\rfloor$ is also wrong, as it can be seen by picking $n=2^i$ for any $i \in \mathbb{N}^+$.