What is the order of $2$ in $(\mathbb{Z}/n\mathbb{Z})^\times$? Is it there some theorem that makes a statement about the order of $2$ in the multiplicative group of integers modulo $n$ for general $n>2$? 

In general it is hard to say something about the order of an integer $a$ modulo $n$ (such that $a\neq -1$ or a perfect square) as $n$ varies. You may be interested to read about Artin's conjecture on primitive roots.
Essentially copied and pasted from the wikipedia article:

For example, take $a = 2$. The conjecture claims that the set of
  primes $p$ for which $2$ is a primitive root (i.e., those primes such that $2$ has the largest possible order modulo $p$, that is, $\text{ord}_p(2)= p-1$) has density
  $C_\text{Artin}=0.373955\ldots$. The set of such primes is (sequence A001122 in OEIS) $$S(2) =
 \{3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139,
 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373,
 379, 389, 419, 421, 443, 461, 467, 491, ...\}.$$ It has $38$ elements
  smaller than $500$ and there are $95$ primes smaller than $500$. The ratio
  (which conjecturally tends to $C_\text{Artin}$) is $38/95 = 2/5 = 0.4$.

