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 $n>2$? 

As others have pointed out, this is not trivial in the slightest. There are a few things you can find out, as  Hagen von Eitzen pointed out in the comments. I created the following plots for $n\in 1\ldots 1000,~~~~~ 1\ldots 10000, ~~~~~1\ldots 100000$. 



There are certain things from this picture that are easy to make sense of, notably the highest line, which is $y = x-1$ and then the other prominent lines for small divisors of $n-1$. I am intrigued by the few points that lie between $n-1$ and $\frac{n-1}{2}$. 
The first few elements of this appear to be:
$$9,25,27,81,121,125,169,243,361,625,729,841...$$
Which are all powers of prime numbers, which makes sense. This is actually A108989, and these are those non-prime numbers $n$ for which the multiplicative order of $2$ modulo $n$ is $\phi(n)$. 
I've also tallied the most common orders for the first $1,000,000$ odd integers and the first few were: 
$$\{2, 155171\}, \{4, 108716\}, \{8, 73643\}, \{12, 67834\}, \{6, 57771\}, \{1, 
  55868\}, \{16, 46866\}, \{24, 45317\}, \{48, 31026\}, \{32, 23299\}, \{36, 
  20669\}, \{96, 16278\}, \{72, 15858\}, \{20, 14476\}, \{10, 13413\}$$
See what you can make of that, it isn't in the OEIS. 
