Why $¬¬\bot \not\in PROP$?

As the author himself alerts to: look at the brackets. What's in $PROP$ is $\left(\neg \left(\neg\bot\right)\right)$ (note the two pairs of $()$).

It isn't true that $\neg \left(\neg \bot\right)=\neg \neg \bot$ simply because they aren't the same string of characters. In mathematical logic, particularly in formal systems, one looks at propositions as actual mathematical objects, more precisely as finite strings of characters. The second character in $\neg \left(\neg \bot\right)$ is $\color{blue}($ which differs from the second character in $\neg \neg \bot$. Therefore they are different.

It is likely that eventually, after some familiarity with these concepts has been acquired,that $\neg \neg \bot$ might be defined as $\left(\neg (\neg \bot)\right)$, but for now they are different. Not only that, as the author proves, $\neg \neg\bot \not \in PROP$ according to the formation rules.

Regarding your example, if you look at $1+1$ and $2$, not as representing objects, but as objects themselves, then they are different because, well..., they are different.

Regarding your example :

$1+1=2$

if you want to "test it" according to van Dalen's rules, you must first take into account that it is a formula of predicate logic and, in particular, of formal number theory.

To show that it is well-formed, you must review the formation rules of the language; see Dirk van Dalen, Logic and Structure (5th ed - 2013), page 56-57 :

Definition 3.3.1 TERM is the smallest set $X$ with the properties [...]

Definition 3.3.2 FORM is the smallest set $X$ with the properties [...]

Then, see page 82 : The language of arithmetic, and the notations introduced in page 83.